The structure of invariants in conformal mechanics

We investigate the integrals of motion of general conformal mechanical systems with and without confining harmonic potential as well as of the related angular subsystems, by employing the SL(2,R) algebra and its representations. In particular, via the tensor product of two representations we construct new integrals of motion from old ones. Furthermore, the temporally periodic observables (including the integrals) of the angular subsystem are explicitly related to those of the full system in a confining harmonic potential. The techniques are illustrated for the rational Calogero models and their angular subsystems, where they generalize known methods for obtaining conserved charges beyond the Liouville ones.


I. INTRODUCTION
Arguably the most important one-dimensional multi-particle system is defined by the inverse-square two-body interaction potential. It has been introduced by Calogero four decades ago and is integrable both with and without a confining harmonic potential [1,2]: (1.1) (1.2) This system continues to attract much interest due to its rich internal structure and numerous applications. So far, various integrable extensions have been constructed and studied, in particular, for trigonometric potentials [3], for particles with spins [4], for supersymmetric systems [5], and for other Lie algebras [6]. Such systems exhibit a rich spectrum of physical properties: fractional statistics [7], Laughlin-type wavefunctions [8] and a resonance valence-bond ground state for a related spin chain [9]. Calogero models appear in many areas of physics and mathematics, like black holes [10], quantum hydrodynamics, or orthogonal polynomials. The rational Calogero models are maximally superintegrable, i.e. they possess N −1 additional integrals of motion. For the classical Hamiltonian without confining harmonic potential, these have been constructed explicitly by Wojciechowski [11]. Later this construction was extended to the quantum case [12,13] and to the inclusion of a confining harmonic potential, where oscillatory behavior with commensurate frequencies implies the superintegrability [14][15][16]. This property has been established also for the hyperbolic Calogero model [17] and the relativistic extension known as the rational Ruijsenaars-Schneider model [18,19].
An important feature of rational Calogero models is the dynamical conformal SL(2,R) symmetry, generated by the Hamiltonian (1.2) together with the dilatation and conformal boost generators [20] Many properties of these systems, like superintegrability [16,17], equivalence to a free-particle system [21,22], or the existence of action-angle variables [23] are simple consequences of the conformal symmetry. The Casimir element of this algebra, coincides with the angular part of the Calogero model and is an integral of motion of both Hamiltonians (1.1) and (1.2). It does not belong to the usual system of Liouville integrals, but its commutator with them produces all additional integrals of motion responsible for the superintegrability. The angular part (1.5) can be considered as a separate (super)integrable system describing a particle moving on the (N −1)-dimensional sphere, which has been defined and studied in a number of recent papers [24][25][26][27][28].
The rational Calogero models are integrable members of the more general class of conformal mechanical systems, whose action is invariant under the conformal transformations (1.3) and which were first introduced in [29]. As a recent application, such systems can describe particle dynamics near the horizon of an extremal black hole [30][31][32].
A lot of what is derived in this paper also applies to any conformal mechanics system. The article is structured in the following way.
In Section 2, we describe the SL(2,R) representation content of integrals of motion in conformal classical mechanics and employ the conformal algebra to expand the system of conserved charges. Any such integral of motion with a definite value of the conformal spin is, by definition, a highest-weight state and thus generates a representation of the conformal algebra. The descendant states in this representation are not conserved, but can be used to construct additional integrals [11] which, of course, are highest weights in another representation. The tensor product of two representations is a convenient way to generate new ones. Thus, given two integrals of conformal classical mechanics, we may decompose the tensor product of their conformal representations into irreducible pieces and pick from each of these the highest state, which will yield a new conserved charge. In this way, extending a method applied in [11,13], we express new integrals of conformal classical mechanics in terms of descendants of two given integrals of motion. The simplest application to the rational Calogero model recovers Wojciechowski's construction [11]. From the standard tensor product, the descendant states are combined symmetrically, i.e. the corresponding phase-space functions are pointwise multiplied. When instead one combines them under the Poisson bracket, it seems that merely the known Liouville integrals are reproduced, but this option needs a more detailed study.
In Section 3, the aforementioned construction of additional integrals of motion is extended to the quantum case. The new integrals appear in symmetrized products of operator-valued conformal representations, which in the semiclassical limit reduce to pointwise products. Section 4 is devoted to the integrals of motion for conformal mechanics in a confined harmonic potential, e.g. (1.1), and to those for the related angular mechanics (1.5). Recently, it has been shown that the spectrum and eigenstates of these two systems are closely related for the quantum Calogero model [28]. Here we study this relation at the level of more general conformal mechanical systems. Starting with one or more integrals of motion for some unconfined conformal mechanics, e.g. (1.2), a system of oscillating observables is constructed for the corresponding angular mechanics. The frequencies of these observables are integer multiples of the basic frequency, the latter being the square root √ I of the angular Hamiltonian [24,26]. These observables can easily be combined to products with vanishing frequency, giving rise to further integrals of motion. Similarly, for the model with a confining harmonic potential, oscillating observables and integrals of motion are derived from conserved charges of the unconfined system, but now the basic frequency is the frequency ω of the confining potential [13]. In both cases we employ appropriate SL(2,R) rotation operators. For the angular system, in addition a noncanonical special conformal transformation inverting the radial coordinate is involved. As a result, we have found the exact relation between the oscillating observables of confined conformal mechanics and of the related angular system. Section 5 revisits the matrix-model construction of the Calogero Hamiltonians H 0 and H ω . We consider the additional integrals described in Section 2 and generated from the standard Liouville integrals of the Calogero Hamiltonian H 0 . Again based on these Liouville integrals, we construct the oscillating observables of the confined system H ω treated in Section 4 and describe them in terms of oscillating matrices. We then prove that the Poisson action of the angular Hamiltonian I on the standard Liouville integrals of H ω produces N −1 additional integrals, which combine with the N Liouville integrals to a complete and independent system. This generalizes a similar property for the unconfined Calogero Hamiltonian H 0 [24,26]. (2.1) The assignment f →f is a Lie algebra homomorphism, the constants on the phase space form its kernel. For an interaction potential V , the vector fieldŝ satisfy the sl(2, R) algebra (1.3), and the vector field of the Casimir elementÎ, of course, commutes with them. Consider now the general conformal mechanical system with the Hamiltonian H 0 obeying the symmetry relations (1.3). First, this implies that the Casimir element (1.5) is an integral of motion of the system with the zero conformal dimension. Next, suppose that the Hamiltonian apart from itself and (1.5) possesses other integrals of motion. Here we plan to study in detail the interrelation of the conformal symmetry and these integrals.
Note that any constant of motion is a highest-weight vector of the conformal algebra (1.3), since it is annihilated by the Hamiltonian. Without any restriction, one can choose it to have a certain conformal dimension (spin), where we have introduced more conventional notations for the raising, lowering and diagonal generators of SL(2,R): The covariant basis for the conformal algebra reads

7)
where the indices are raised and lowered by the metric g αβ = diag(−1, 1, −1). The Casimir elementŜ 2 of the related vector field algebra equals −s(s+1) times the identity on an sl(2, R) representation of spin s as given in (2.3). It is important to distinguish between the square of the conformal spin (2.9), as a second-order differential operator, and the vector fieldÎ generated by the Casimir invariant (2.8), as a first-order operator: Note thatÎ does not preserve representations but acts as an intertwiner between them. The descendants form the basic states of the spin-s representation of the conformal algebra (2.9). For generic real values of s, there is an infinity of them. For non-negative integer or half-integer values of s however, the state I s,2s+1 either vanishes or it is another integral of motion for the conformal mechanics Hamiltonian.
If I s,2s+1 = 0, then we deal with a finite-dimensional irreducible representation. This includes the rational Calogero model, whose Liouville constants of motion are polynomials of order (2s) in the momenta. Their multiplets are nonunitary and similar to the spin-s representations of su (2).
If I s,2s+1 does not vanish, it is another integral of the conformal Hamiltonian, since the spin raising operatorŜ + annihilates it, as is easy to check using the definition (2.3) and commutation relations (2.5). As a highest-weight state, it generates another conformal representation, which forms an invariant subspace. We thus encounter an indecomposable representation, which is reducible but not fully reducible.
The time evolution of the observables (2.11) is given by a kth order polynomial in time [20], since the (k+1)th power of the evolution operator d/dt =Ĥ 0 =Ŝ + annihilates it. However, they can be used to construct new integrals of motion different from I s . This construction is be done in terms of the representation theory of the conformal algebra, as will be described below.
Denote by (s) the sl(2, R) representation (2.11), generated by the integral of motion I s . For two integrals I s1 and I s2 the products of the corresponding descendant states form the product representation, which decomposes into a direct sum of representations: For finite dimensional irreducible representations, this series terminates at (|s 1 −s 2 |), giving rise to the usual momentum sum rule in quantum mechanics. The highest-weight states of the kth multiplet in the decomposition (2.12) are also integrals of motion with conformal spins s = s 1 + s 2 − k, which we denote by I (s1,s2) s . They can be calculated using the sl(2, R) Clebsch-Gordan coefficients. However, it is easier to derive them directly with the commutation relation and the highest-state conditions (2.3). Choosing a suitable normalization factor, we define the new integrals of motion as For generic real values of s 1 or s 2 , we obtain an infinity of these. For integer or half-integer values of the spins however, there appears only a finite number, limited by k < 2min(s 1 , s 2 ). In multi-particle models, the new integrals I (s1,s2) s are quadratic in the descendants and hence involve a double sum over the particle index, making them composite objects. Let us consider particular cases of the composite integrals of motion (2.14). The case k = 0 is uninteresting, since it merely yields the product I s1 I s2 . For k = 1 we have which is a new integral of motion for any pair I s1 and I s2 [11]. If the first integral is the Hamiltonian, I s1 = I 1 = S + , then we simply obtain the bracket with the Casimir element (2.8), In the last equation the second relation in (2.10) is used. For the N -particle Calogero system, the Casimir invariant I of the conformal algebra produces in this way the additional N −1 integrals of motion from the Liouville integrals [24,26]. Another special case is s 1 = s 2 . Exchanging l ↔ k − l in the sum (2.14), one concludes that terms with k odd cancel out, leaving only even values for k. Rather than simply multiplying the descendants on the right-hand side of (2.14), one may take their Poisson bracket instead, since they are phase-space functions. The conformal tensor product decomposition is unaffected, and the Clebsch-Gordan coefficients remains the same, For the Calogero model, the Poisson bracket of two conformal representations generated by Liouville integrals I s1 and I s2 , was considered already in [33]. For the standard Liouville integrals in the simplest cases the above formula yields nothing new as we shall sketch in Section 5. Therefore, we further discuss only the pointwise products (2.14).

III. SL(2,R) STRUCTURE OF THE INTEGRALS OF MOTION: QUANTUM CASE
In passing from the classical to the quantum model, we replace The expressions (1.1), (1.2) and (1.4) for the Hamiltonians and conformal group generators remain the same, except that symmetric (Weyl) ordering between momenta and coordinates must be used, which affects the dilatation The hermitian generators obey the quantum commutation relations The expressions (2.4) and (2.6) for the invariant conformal generators S ±,z,x,y remain unchanged, while the quantum analogue of (2.7) reads Note that, in contrast to the well known su(2) raising and lowering operators, the sl(2, R) operators S ± are hermitian and thus not mutually conjugate. The Weyl ordering becomes essential in the Casimir element Any quantum observable f defines an infinitesimal evolution map given by the operator which is the quantum analog of the classical vector field (2.1) and reduces to it in the semiclassical limit. Again, the assignment f →f is a Lie algebra homomorphism. In this way, we get a (not necessarily unitary) representation of the conformal algebra on the space of quantum operators. It was introduced and used for the construction of additional integrals of the quantum Calogero system [13], simplifying an earlier procedure [12].
In the adjoint action (3.7), the quantum commutation relations of the conformal group generators coincide with their classical commutators (2.5) and (2.9).
As in the classical case, any spin-s quantum integral of motion (2.3) generates a highest-weight representation (2.11) of the conformal algebra. The product of two such representations is subject to the sum rule (2.12), and the highest states (2.14) yield new integrals of motion for the quantum conformal Hamiltonian. However, since quantum physical observables are supposed to be hermitian, the expressions for I (s1,s2) s1+s2−k must be self-conjugate. In order to achieve this, it suffices to symmetrize the products of descendants in I (s1,s2) s , i.e. [39] s1+s2−k is simply the irreducible component of the symmetrized tensor product (I s1 ⊗ I s2 ) + : In the classical limit → 0, it reduces to (2.14). The first nontrivial case corresponds to k = 1: which is the quantum version of the classical integrals (2.15) [12,13] and produces, for s 1 = 1 and I 1 = S + , as the simplest quantum integrals beyond the Liouville ones.

IV. RELATING ANGULAR MECHANICS AND CONFINED CONFORMAL MECHANICS
The angular part I of the Hamiltonian coincides with the Casimir element S 2 of the conformal algebra (2.8). It defines a mechanical subsystem depending only on the angular coordinates and momenta u, to which we refer as angular mechanics. Its integrals of motion have been studied for general conformal mechanics [24,26] and, in particular, for the Calogero model [27]. Motion in the angular subsystem (1.5) is naturally bounded, as it is for the harmonically confined conformal system (1.1). Confined integrable systems feature quantities which oscillate in time with a fixed frequency. Examples are the angle variables, which, together with their canonically conjugated action variables, the Liouville integrals, parametrize the phase space of the system. In the case of commensurate frequencies, additional integrals exist and are expressed completely via the angles [34]. This property is known as superintegrability. Note that the existence of such integrals does not require integrability: two quantities oscillating with commensurate frequencies are sufficient. In this section we study consequences of the existence of higher-order integrals in two confined mechanical systems, namely in angular mechanics and in harmonically confined conformal mechanics. For both we will construct quantities which oscillate in time with integral multiples of a basic frequency. These frequencies are proportional to the spin projections in a finite-dimensional SL(2,R) representation, which is realized differently in the two systems.
Let us first focus on the angular mechanics case. In order to construct the angular subsystem of a conformal mechanics model, it is suitable to express the conformal generators (2.4) and (2.2) in terms of angular coordinates and momenta, u = (θ α , p θα ), and of radial ones, Using (4.2) and (4.3), it is easy to see [26] that the highest-weight condition (2.3) is equivalent tô where we introduced the one-dimensional vector fields [40] S R which form another sl(2, R) algebra. Note thatÎ acts only on the angular variables while theŜ a feel just the radial dependence. Therefore, (4.4) relates the angular dependence of I s to its radial one. For vanishing angular part, the above new generators are dual to the conformal generators (4.3): where the duality map inverts the radial coordinate but leaves the angular ones unchanged. Evidently, it is not a canonical transformation, but the dual generators (4.5) obey the same algebraic relations (2.5) as the standard generators (4.3) do. For 2s being integer, any spin-s integral, as defined in (2.3), can be decomposed into terms with a homogeneous radial dependence separated from the angular one [41], The radial functions p 2s−l r /r l form a basis of the spin s-representation of the sl(2, R) algebra (4.5), whereŜ R z is diagonal. The inversion (4.6) maps them to the equivalent representation given by the S a | I=0 acting on the polynomials p 2s−l r r l of order 2s. The conformal descendants of the integrals satisfy the following decomposition, In particular, the lowest descendant (k = 2s) reduces to a single term independent of the radial momentum, so its radial and angular dependencies factorize: I s,2s (p r , r, u) = (2s)! r −2s f s,0 (u). (4.10) This is reminiscent of the wavefunctions of the quantum Calogero Hamiltonian [28]. The transformation (4.9) from {f s,2s−k } to {I s,k } is given by a triangular matrix with diagonal elements (k)! r −2s . Therefore, it is invertible, and f s,l can be expressed in terms of {I s,2s , . . . , I s,2s−l }. It is convenient to pass from the radial variables 1/r and p r to the complex combinations As any linear map, it extends to the (2s+1)-dimensional space of polynomials of degree 2s, where U s is a 2s × 2s matrix depending on √ I. Its elements can be derived from those of the fundamental representation, ( U 1 2 ) kl , which are determined by (4.11). In the new basis, the dualized spin operators take the form The decomposition (4.8) of a conformal mechanics integral in the new monomial basis (4.12) defines shifted angular harmonicsf s,l (u) via (4.14) We remark that our bases are not normalized. The standard, normalized, sl(2, R) representation basis [26] is obtained by using [26]  According to the last equation in (4.13), the action (4.4) of the angular Hamiltonian becomes diagonal in the new coordinates. Hence, its action on the decomposition (4.14) implies that, for I > 0, the shifted harmonics oscillate with the frequencies equal to integral multiples of √ I: The basic frequency √ I is, of course, a constant of motion. The map (4.12) defined by the matrix U s can be expressed s-independently in terms of the conformal generators, In the fundamental representation s = 1/2, the operatorsŜ R y andŜ R z are represented in terms of Pauli matrices as i 2 σ y and 1 2 σ z , respectively, but the expression naturally extends to the polynomial spin-s representation of the conformal algebra (4.13). On the basis (4.12) or (4.15), it evidently reduces to the matrix U s from (4.14). From (4.18) it is clear that the latter can be expressed in terms of Wigner's small d-matrix, which is explicitly done in Appendix A.
With the help of (A.7) and (4.16), the relation between the normalized original and shifted angular harmonics take the following form,f Using the definition (2.6) and the relation q −ŜzŜ ± qŜ z = q ∓1Ŝ ± for q = i √ I, which is a direct consequence of the commutation relations (2.5), the adjoint action of the operator (4.18) on the generators of the conformal algebra (4.5) or (4.13) can be calculated: We emphasize again that the operator (4.18) is not canonical since the vector fieldsŜ R z andŜ R y are not Hamiltonian. The expression (4.21) is, in general, complex and multi-valued. When the potential is positive, as is the case in Calogero models, the angular part is strictly positive and the operator (4.18) is complex but single-valued. In any case, all square roots will cancel in the final expressions for the constants of motion.
The second part of this section deals with conformal mechanics in an external harmonic potential. We shall see that, again, the integrals of motion are derived from the descendants I s,k with the help of a Wigner rotation, similar to the angular mechanics case above. Adding a harmonic confining potential is a deformation compatible with the conformal symmetry: The last relation has the same structure as in (4.4), and it is mapped to the latter expression under the substitution links the Hamiltonian with harmonic potential to the diagonal conformal generator, The operator U defines a complex-valued nonlocal map which, in contrast to its counterpart U , is canonical. Nevertheless, its action on the space spanned by the descendants (2.11) is given by an SL(2,C) representation matrix. The SL(2,C) transformation (4.24) mapping S z to H ω is determined only up to an overall z rotation from the right [42]. Our choice of (4.24) is fixed by the transformation rules The operator (4.24) diagonalizes also the basis (2.11) of the spin-s representation comprising the descendants of the spin-s integral of conformal mechanics. In complete analogy with (4.12), we definẽ I s,l := (iω) s 2s k=0 I s,k (U s ) kl , (4.27) where the matrix elements (U s ) kl depend on ω. Applying the second relation in (4.25) to these shifted basis states, we getĤ ωĨs,l = −2imωĨ s,l with m = s−l. (4.28) Due to these eigenvalue relations, the "harmonics" (4.33) oscillate in time with integer frequencies proportional to the spin projection value,Ĩ sm (t) = e −2imω(t−t0)Ĩ sm (t 0 ). (4.29) Here and in the following, we use the standard basis for sl(2, R) representations by introducing which is distinguished from the previous basis by omitting the comma between indices. In this basis, the decomposition (4.9) into radial and angular parts reads where we applied the same index nomenclature to the f s given by (4.16).
As was mentioned before, the transformation U is canonical and preserves the Poisson brackets, In particular, if someĨ sm are in involution, then the corresponding I sm are in involution, too. The matrix form of the action of the shift operators (4.24) and (4.18) acquires the following form, These formulae are analogous to (4.19) and (4.20). Actually, the transformation (4.33) is equivalent to the transformation (4.12), which is the inverse transpose of (4.19), according to the definition (4.14). Finally, we would like to take advantage of the structural analogy of the two models and directly relate the corresponding shifted harmonics (4.19) and (4.33). To this end, we first substitute (4.20) into (4.31) and then insert the resulting expression for I sm into (4.33). Ultimately, we arrive at which expresses the harmonic functions of the confined conformal mechanics in terms of the harmonics of the related angular system. Here, the prime over the sum restricts the indices by the condition m 1 +m 2 ≥ 0, so that all arguments of the binomial coefficients in are positive.

V. THE RATIONAL CALOGERO MODEL WITH HARMONIC POTENTIAL
In this section we specialize to the Calogero model and employ the well known matrix-model description. Firstly, for the unconfined Calogero Hamiltonian H 0 (1.2), we work out the explicit form of the integrals of motion (2.14) composed of the conformal descendants of two standard Liouville integrals. Secondly, for the confined Calogero Hamiltonian H ω (1.1), we present a simple expression for the oscillating observableĨ sm of Section 4. Thirdly, we act with the Hamiltonian vector fieldÎ related to the angular Hamiltonian on the N standard Liouville integrals of H ω and obtain the additional N −1 integrals for this model. The functional independence of all 2N −1 integrals is proven explicitly, demonstrating that they comprise a complete system. A similar property is already known for H 0 [24,26]. The Hamiltonians (1.1) and (1.2) can be obtained by SU(N ) reduction, respectively, from the hermitian matrix models [15,35,36] [43] Here, P and Q are hermitian matrices containing the canonical momenta and coordinates, subject to The round brackets denote the SU(N ) trace, The matrix Hamiltonians (5.1) describe a homogeneous N 2 -dimensional oscillator and a free particle in R N 2 , respectively. In the matrix-model reduction, the Calogero coupling g is recovered by the gauge-fixing relation 2) provide the solution of above equation [15,36]: Furthermore, the standard Liouville integrals and their descendants (2.11) of the Calogero Hamiltonian H 0 are obtained from the reduction of [33]: respectively. Here, s ≤ N/2, and the index 'sym' means symmetrization over all orderings of P, Q matrices inside the trace, e.g.
(P n Q) sym = (P n Q) , The symmetrized traces can be computed by means of the generating function which can be considered as an extension of Newton's binomial formula.
Using (5.7), we can write down the integral (2.14) in terms of symmetrized traces, The coefficient in front of the sum is not essential and can be ignored. Substituting s 1 = 1, s 2 = n/2 and k = 1 in above equation, we arrive at the integral proportional to {I, I n 2 }, as was shown in (2.16), For n = 2 it vanishes, but the remaining N −1 integrals together with the Liouville ones (P n ), 1 ≤ n ≤ N , constitute a complete system of constants of motion of H 0 . The functional independence of this set can be seen from the freeparticle limit g = 0 where, according to (5.5), the matrices P , Q are diagonal. This property reveals the role of the angular part: it generates the full system of integrals for H 0 by acting on the Liouville ones. Below we will show that this role of I extends to the confined system H ω as well. For the particular case of s 1 = s 2 = n/2 and k = 2 in (5.10), we find the integral At n = 2 we recover the Casimir element of (5.6) describing the angular mechanics. Before passing to the confined model H ω , we briefly consider the issue of the H 0 integrals (2.17) obtained by taking Poisson brackets. Let us take Liouville integrals (5.7) for I s1 and I s2 . In the k = 0 special case, (2.17) reduces to {I s1 , I s2 } = 0, since I s1 and I s2 are in involution. For k = 1 one finds the Liouville integral I s1+s2−1 . The k = 2 case vanishes again, as can be calculated using (5.2) and (5.7). In general, the relations (5.5) imply that the Poisson brackets in (2.17) evaluate to {I s1,k1 , I s2,k2 } ∼ (s 1 k 2 − s 2 k 1 )I s1+s2−1,k1+k2−1 + . . . , (5.13) where the remaining terms are of order O(g) and thus vanish in the free-particle limit. Their structure is more complicated: for higher spins they may contain, besides traces, also mean values e| . . . |e of products of P, Q matrices. In total, one obtains the whole infinite algebra of observables of the Calogero model [37]. For studying the Calogero system in an external harmonic potential, it is most suitable to employ the creation and annihilation combinations In terms of these, the Hamiltonian H ω reads [35] H The matrix variables A ± oscillate in time with frequency ω: Using the canonical brackets in (5.15) and the expressions (5.6) for the SL(2,R) generators, one can calculate the action of the transformation (4.24) on the phase-space variables of the matrix model: Recall that this transformation maps the diagonal generator S z to the Hamiltonian H ω according to (4.25). It turns into the analogous transformation U given by (4.11) upon substituting (P, Q) → (p r , r −1 ), In this context, the analogy between confined between confined Calogero model and related angular system becomes more transparent in the matrix model description. It expands also to the homogeneous polynomials, which form spin-s representations of conformal algebras According to (4.30) and (5.7), the matrix form of original and shifter normalised states are: According to (4.29),Ĩ sm oscillates with the frequency −2mω, which also can be seen from (5.16).
In fact, the trace of any product of A ± matrices, oscillates with integer frequency equal to ω i σ i . Any product of such observables with a vanishing total sum of the σ i will be an integral of motion of the Calogero system (1.1). The number of such integrals is significantly higher than 2N −1, but they are not independent. The Liouville integrals of the Hamiltonian H ω can be extracted from this general set using either the Lax-pair method or the symmetries of the original matrix model (5.1) or (5.15). We recall the second way described in the review [35]. In terms of the hermitian left-multiplication generator or Lax matrix A + A − , one has The related flow is not symplectic, because the left (or right) multiplication does not preserve the Poisson brackets in (5.15), as the adjoint action does. The matrix elements of A + A − obey the U(N ) commutation relations, and the Liouville integrals (5.21) can be identified with the Casimir elements of that group [35]. The Liouville integralsĨ n are unsymmetrized analogs of the integralsĨ s=n m=0 (5.20) with integer spin s = n ∈ N. They form another set of Liouville integrals [16]. Up to a numerical factor, the integralsĨ n0 andĨ n coincide in their term of highest power in the momenta, corresponding to the free-particle limit g = 0. Note that, according to (4.32), the descendants I n0 also are in involution. The first integral is proportional to the Hamiltonian and corresponds to the central U(1) part,Ĩ The remaining two bilinear traces of (5.14) are also related to the conformal generators. Using their matrix form (5.6) and (2.4), we obtain: ¿From these equations it is easy to express the angular Hamiltonian, described by the Casimir element (1.5) of the conformal algebra, in terms of A ± matrices. It has a rather simple form: Since J 1 = 0, we find 2N −1 integrals (Ĩ 1 , . . . ,Ĩ N , J 2 , . . . J N ), which appear to form a complete set of integrals for H ω . Their functional independence is shown in Appendix B. This confirms the superintegrability of the Calogero model in the harmonic potential. The construction of the additional integrals J n is similar to the one for the free Calogero Hamiltonian described by equation (5.11) above.
The functional independence of this set of integrals is equivalent to the nondegeneracy of the Jacobian matrix where M ij denotes a minor of the matrixB.
In the simplest case of equal phases ϕ i = ϕ, this determinant is given by characteristic polynomial of the rank-one matrixb ij = ρ j with the value ρ = N i=1 ρ i for the eigenvalue variable, which can be easy calculated: (B.11) Therefore, the Jacobian vanishes at special points only. The matrixB generically depends analytically on the arguments ϕ i , so its determinant can vanish only at a set of measure zero in the space of ϕ i .