The Novel Lie-Algebraic Approach to Studying Integrable Heavenly Type Multi-Dimensional Dynamical Systems

The review is devoted to a novel Lie-algebraic approach to studying integrable heavenly type multi-dimensional dynamical systems and its relationships to old and recent investigations of the classical Buhl problem of describing compatible linear vector field equations, its general Pfeiffer and modern Lax-Sato type special solutions. Eespecially we analyze the related Lie-algebra structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Pleban′ski and later analyzed in a series of articles. The AKS-algebraic and related -structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie–Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is shown that all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a very interesting Lagrange–d’Alembert type mechanical interpretation of the devised integrability scheme with the Lax–Sato equations is also discussed.


Introduction
In 1928 the French mathematician Buhl in his works [1,2] posed the problem of classifying all infinitesimal symmetries of a given linear vector field equation
n j a C j n ∈   It is easy to show that the problem under regard is reduced [3] to describing all possible vector fields for all x∈  n and = 1, . k n This M.A. Buhl problem above was completely solved in 1931 by the Ukrainian mathematician Pfeiffer in the works [4][5][6][7][8]; where he has constructed explicitly the searched set of independent vector fields (3); having made use effectively of the full set of invariants for the vector field (2) and the related solution set structure of the Jacobi-Mayer system of equations; naturally following from (4). Some results; yet not complete; were also obtained by Popovici [9]. Some years ago the Buhl MA type equivalent problem was independently reanalyzed once more by Japanese mathematicians Takasaki and Takebe [10,11] and later by Bogdanov et al. [12] for a very special case when the vector field operator (2) depends analytically on a "spectral" parameter λ∈: Based on the before developed Sato theory [13,14]; the authors mentioned above have shown for some special kinds of vector fields (5) that there exists an infinite hierarchy of the symmetry vector fields for all k,j∈ + . Moreover; in the cases under regard; the compatibility conditions (7) proved to be equivalent to some very important for applications heavenly type dispersionless equations in partial derivatives.
In the present work we investigate the Lax-Sato compatible systems; the related Lie-algebraic structures and complete integrability ∈    and some Casimir invariants; we have successively demonstrated that their compatibility condition coincides exactly with the corresponding heavenly equations under consideration.
As a by-product of the construction; devised recently in works [27,28]; we prove that all the heavenly equations have a similar origin and can be represented as a Lax compatibility condition for special loop vector fields on the torus  n . We analyze the structure of the infinite hierarchy of conservations laws; related to the heavenly equations; and demonstrate their analytical structure connected with the Casimir invariants is generated by the Lie-Poisson structure on . *   Moreover; we have extended the initial Lie-algebraic structure for the case when the the basic Lie algebra  �  ( ) n diff  is replaced by the adjacent holomorphic Lie algebra : =   of vector fields on  ×  n . Typical examples are presented for all cases of the heavenly equations and it is shown in detail and their integrability is demonstrated using the scheme devised here. This scheme makes it possible to construct a very natural derivation of well known Lax-Sato representation for an infinite hierarchy of heavenly equations; related to the canonical Lie-Poisson structure on the adjoint space . *  We also briefly discuss the Lagrangian representation of these equations following from their Hamiltonicity with respect to both intimately related commuting evolutionary flows; and the related bi-Hamiltonian structure as well as the Bäcklund transformations. As a matter of fact; there are only a few examples of multi-dimensional integrable systems for which such a detailed description of their mathematical structure has been given. As was aptly mentioned [29]; the heavenly equations comprise an important class of such integrable systems. This is due in part to the fact that some of them are obtained by a reduction of the Einstein equations with Euclidean (and neutral) signature for (anti-) self-dual gravity; which includes the theory of gravitational instantons. This and other cases of important applications of multi-dimensional integrable equations strongly motivated us to study this class of equations and the related mathematical structures. As a very interesting aspect of our approach to describing integrability of the heavenly dynamical systems; there is a very interesting Lagrange-d' Alembert type mechanical interpretation. We need to underline here that the main motivating idea behind this work was based both on the paper by Kulish [30]; devoted to studying the super-conformal Korteweg-de-Vries equation as an integrable Hamiltonian flow on the adjoint space to the holomorphic loop Lie superalgebra of super-conformal vector fields on the circle; and on the insightful investigation by Mikhalev [31]; which studied Hamiltonian structures on the adjoint space to the holomorphic loop Lie algebra of smooth vector fields on the circle. We were also impressed by deep technical results [10,11] of Takasaki and Takebe; who fully realized the vector field scheme of the Lax-Sato theory. Additionally; we were strongly influenced both by the works of Pavlov; Bogdanov; Dryuma; Konopelchenko and Manakov [12,[32][33][34]; as well as by the work of Ferapontov and Moss [35]; in which they devised new effective differential-geometric and analytical methods for studying an integrable degenerate multi-dimensional dispersionless heavenly type hierarchy of equations; the mathematical importance of which is still far from being properly appreciated. Concerning other Lie-algebraic approaches to constructing integrable heavenly equations; we mention work by Szablikowski and Sergyeyev [36,37]; Ovsienko [17,18] and by Kruglikov and Morozov [38].
We present interesting examples of the Lie-algebraic description of typical integrable heavenly equations amongst which the Mikhalev-Pavlov equation [31]; the first and second reduced Shabat type [39] and Hirota heavenly equations [40], the Liouville type [33] equations and some other.

A vector field on the torus and its invariants
Consider a simple vector field X:  n T(× n ) on the (n+1)dimensional toroidal manifold  n for arbitrary n∈ + , which we will write in the slightly special form for some function ∈C 2 ( n ;), which we will call an "invariant" of the vector field.
Next; we study the existence and number of such functionallyindependent invariants to eqn. (9). For this let us pose the following Cauchy problem for eqn. (9): Find a function ψ∈C 2 ( n ;), which at point t (0) ∈ satisfies the condition  C ψ ∈   For eqn. (9) there is naturally related parametric vector field on the torus  n in the form of the ordinary vector differential equation to which there corresponds the following Cauchy problem: find a function x:→ n satisfying for an arbitrary constant vector z∈ n . Assuming that the vectorfunction aC 1 (× n ;R n ); it follows from the classical Cauchy theorem [41] on the existence and unicity of the solution to (10) and (11); that we can obtain a unique solution to the vector eqn. (10) as some function Φ∈C 1 (× n ; n ),x=Φ(t,z), such that the matrix ∂Φ(t,z)∂z is nondegenerate for all t∈ sufficiently close to t (0) ∈. Hence; the Implicit Function Theorem [42,43] implies that there exists a mapping Ψ: n  n , such that for every z∈ n and all t∈ sufficiently enough to t (0) ∈. Supposing now that the functional vector is constructed; from the arbitrariness of the parameter z∈ n one can deduce that all functions ( ) : , n are functionally independent invariants of the vector field eqn. (9); that is ( ) = 0, = 1, .
Thus; the vector field eqn. (9) has exactly n∈ + functionally independent invariants; which make it possible; in particular; to solve the Cauchy problem posed above. Namely; let a mapping α : n  be chosen such that for all x∈ n and a fixed t (0) ∈. Inasmuch as the superposition of functions    is; evidently; also an invariant for the eqn. (9); it provides the solution to this Cauchy problem; which we can formulate as the following classical lemma.

Lemma 2.1:
The linear eqn. (9); generated by the vector field (10) on the toroidal manifold × n , has exactly n + functionally independent invariants. Consider now a differential form χ (n) ∈Λ n ( n ), generated by the vector of independent invariants (12); additionally depending parametrically on the vector evolution parameter t n : where; by definition; for any ψ∈C 2 ( n × n ;) the differential As follows from the Frobenius theorem [35,42,44]; the Plucker type form (13) is for t∈ n nonzero on the torus  n owing to the functional independence of the invariants. It is easy to see that the following [9] Jacobi-Mayer type relationship holds on the manifold  n where on the right-hand side one has the volume measure on the torus  n , which is naturally dependent on t∈ n owing to the vector field relationships (10). Taking into account that for all invariants ( ) 2 ( ; ) = 1, their substitution into (15) gives rise; owing to the independence of the differentials , = 1, , s dt s n to the following set of the compatible vector field relationships for any = 1, . s n The latter property; as it was demonstrated by Pfeiffer [9]; makes it possible to solve effectively the M.A. Buhl problem and has interesting applications [13,35] in the theory of completely integrable dynamical systems of heavenly type; which are considered in the next section.

Vector field hierarchies on the torus with "spectral" parameter and the Lax-Sato integrable heavenly dynamical systems
Consider some naturally ordered infinite set of parametric vector fields (8) on the infinite dimensional toroidal manifold  are the evolution parameters; and the dependence of smooth vectors ( ) ( ) 0 ( , ) , , k k n a a k + ∈ × ∈     on the " spectral" parameter λ= is assumed to be holomorphic. Suppose now that the infinite hierarchy of linear equations for k + has exactly n+1∈ + common functionally independent invariants ( ) ( ) j ψ λ ∈ 2 ( ; ), = 0,   on the torus  n , suitably depending on the parameter λ∈. Then; owing to the existence theory [42,43] for ordinary differential equations depending on the " spectral" parameter λ∈, these invariants may be assumed to be such that allow analytical continuation in the parameter λ∈ both inside 1 + ⊂   of some circle  1 ⊂ and subject to the parameter This means that as |λ|→∞ we have the following expansions: where we took into account that where x := ( , ) , is the Jacobi determinant of the on the manifold × n Inasmuch this mapping subject to the parameter λ∈ has analytical continuation [43] inside 1 + ⊂   of the circle  1 ⊂ and subject to the parameter λ −1  as |λ|→∞ outside 1 − ⊂   of this circle  1 ⊂, one can easily obtain from the vanishing differential expressions for all = 1, j n and the relationship (21) on the manifold  n of the independent variables x∈ n , evolving analytically with respect to the parameters ( ) , j k τ ∈  = 1, , , j n k + ∈  the following Lax-Sato criterion: where (…)_ means the asymptotic part of an expression in the bracket; depending on the parameter λ −1  as |λ|→∞. The substitution of expressions (22) into (23) easily yields for all , = 1, . k j n + ∈  These relationships (24) comprise an infinite hierarchy of Lax-Sato compatible [11,12] linear equations; where (…) + denotes the asymptotic part of an expression in the bracket; depending on nonnegative powers of the complex parameter λ∈. As for the independent functional parameters ( ) 2 ( ; )   for all , = 1, , k j n + ∈  one can state their functional independence by taking into account their a priori linear dependence on the independent evolution parameters t k ∈, k∈ + . On the other hand; taking into account the explicit form of the hierarchy of eqn. (24); following [13]; it is not hard to show that the corresponding vector fields on the manifold × n satisfy for all , , , = 1, , k m j l n + ∈  the Lax-Sato compatibility conditions which are equivalent to the independence of the all functional parameters ( ) 1 ( ; ), As a corollary of the analysis above; one can show that the infinite hierarchy of vector fields (18) is a linear combination of the basic vector fields (25) and also satisfies the Lax type compatibility condition (26). Inasmuch the coefficients of vector fields (25) are suitably smooth functions on the manifold , n + ×    the compatibility conditions (26) yield the corresponding sets of differential-algebraic relationships on their coefficients; which have the common infinite set of invariants; thereby comprising an infinite hierarchy of completely integrable so called heavenly nonlinear dynamical systems on the corresponding multidimensional functional manifolds. That is; all of the above can be considered as an introduction to a recently devised [11][12][13]33] constructive algorithm for generating infinite hierarchies of completely integrable nonlinear dynamical systems of heavenly type on functional manifolds of arbitrary dimension. It is worthwhile to stress here that the above constructive algorithm for generating completely integrable nonlinear multidimensional dynamical systems still does not make it possible to directly show they are Hamiltonian and construct other related mathematical structures. This important problem is solved by employing other mathematical theories; for example; the analytical properties of the related loop diffeomorphisms groups generated by the hierarchy of vector fields (18).

Remark 2.2:
The compatibility condition (26) allows an alternative differential-geometric description based on the Liealgebraic properties of the basic vector fields (25). Namely; consider the manifold , n× +   as the base manifold of the vector bundle for an equivalence relation ρ and the (holomorphic in ; naturally acting on the vector space E. The structure group can be endowed with a connection ϒ by means of a mapping generated by the set of parametric vector fields (25); and naturally acting on any mapping 2 ( ( ; )  It is easy now to see that the corresponding to (27) zero curvature condition 2 = 0 h d is equivalent to the set of compatibility eqn. (26). Moreover; the parallel transport equation coincides exactly with the infinite hierarchy of linear vector field eqn. (24); where 2 ( ; ) which is equivalent to the zero curvature condition 2 = 0, h d makes it possible to retrieve [23,45] the corresponding connection ϒ by constructing a mapping in the form (27). These and other interesting related aspects of the integrable heavenly dynamical systems shall be investigated separately elsewhere.

Example: the vector field representation for the Mikhalev-Pavlov heavenly type equation
The Mikhalev-Pavlov equation was first constructed [32] and has the form where u∈C ∞ ( 2 × 1 ;) and (t,y,x)∈ 2 × 1 . Assume now [13] that the following two functions where as the complex parameter λ→∞. By applying to the invariants (30) the criterion (23) in the form one can easily obtain the following compatible linear vector field equations are independent differential-algebraic polynomials in the variable uC ∞ ( 2 × ∞ × 1 ) and algebraic polynomials in the spectral parameter λ∈, calculated from the expressions (24). Moreover; as one can check; the compatibility condition (26) for the first two vector field eqn. (32) yields exactly the Mikhalev-Pavlov eqn. (29).

Example: The Dunajski metric nonlinear equation
The equations for the Dunajski metric [45] are R T One can construct now; by definition; the following asymptotic expansions ∞ are constant parameters. Then the condition (23) yield a compatible hierarchy of the following Lax-Sato type linear vector field equations: are some independent vector-valued differential-algebraic polynomials [33] in the variables and algebraic polynomials in the spectral parameter λ∈, calculated from the expressions (24). In particular; the compatibility condition (26) for the first two equations of (35) is equivalent to the Dunajski metric nonlinear eqn. (33).
The description of the Lax-Sato equations presented above; especially their alternative differential-geometric interpretation (27) and (28); makes it possible to realize that the structure group Diff hol ( n ) should play an important role in unveiling the hidden Liealgebraic nature of the integrable heavenly dynamical systems. This is actually the case; and a detailed analysis is presented in the sequel.

Heavenly Equations: The Lie-Algebraic Integrability Scheme
The corresponding Lie subalgebras for some fixed p,q∈ + . We took above; by definition [29,42]; a loop vector field ( ( )) n a T ∈ Γ    and a loop differential 1-form 1 ( ) n l ∈ Λ    given as used before [46]. The Lie commutator of vector fields , a b   ∈   is calculated the standard way and equals The Lie algebra   naturally splits into the direct sum of two Lie subalgebras = , for which one can identify the dual spaces 1 1 , , Having defined now the projections one can construct a classical -structure [25,26,47] on the Lie algebra   as the endomorphism : , which allows to determine on the vector space   the new Lie algebra structure for any , , a b ∈     satisfying the standard Jacobi identity.
 is a seed element and  are the standard functional gradients at l * ∈    with respect to the metric (37). The related to (45) where for any seed element the gradients and the coadjoint action (46) can be equivalently rewritten; for instance; in the case q=0, as for any = 1, .
j n If one takes two smooth functions ( ) ( ) at any seed element  the following coadjoint action relationships hold: which can be equivalently rewritten (as above in the case q=0) as and similarly where the expressions are true vector fields on  n , yet the expressions are the usual matrix homomorphisms of the Euclidean space  n .
Consider now the following Hamiltonian flows on the space where which holds for all y,t∈ and arbitrary λ∈.
Proof. The compatibility of commuting flows (58) implies that for all y,t∈ and arbitrary λ∈. Taking into account the expressions (57); one has for any vector field From (61) where the degree p∈ + can be taken as arbitrary. Upon substituting (63) into (49) one can find recurrently all the coefficients ∇h(l) j , j + , for some positive integers p y ,p t ∈ + .

Remark 3.2:
As mentioned above; the expansion (63) is effective if a chosen seed element l * ∈    is singular as |λ|→∞. In the case when it is singular as |λ|0, the expression (63) should be replaced by the expansion for an arbitrary p∈ + , and the projected Casimir function gradients then are given by the expressions for some positive integers p y ,p t  + . Then the corresponding flows are; respectively; written as The above results; owing to 3.1; can be formulated as the following main proposition.   depending parametrically on λ,µ∈ and evolution variables (y,t)∈ 2 be such that a seed loop differential form satisfies the invariance condition It generates a Casimir invariant ( ) h I * ∈   for which the expansion (63) as |λ|→∞ is given by the asymptotic series and so on. If further one defines it is easy to verify that As a result of (76) and the commuting flows (68) on *   we retrieve (the equivalent to the Mikhalev-Pavlov [20] ) eqn. (72) vector field compatibility relationships We now study the Bäcklund transformation for two special solutions x t y u x t y u x t y λξ λ − +   or; equivalently; where k∈\{0} and α∈C 2 ( 2 ;) is some mapping. As the loop diffeomorphism (79) should simultaneously satisfy the vector field eqn. (71); giving rise (at α(y,t,λ)=0) to the following Bäcklund type differential relationships which hold for any λ∈ and two special solutions

Example: The Witham heavenly type equation
Consider the following [48][49][50][51][52] as λ→∞ and as λ→0 Based on the expressions (82) and (83); one can construct [29] the following commuting to each other Hamiltonian flows with respect to the evolution parameters y,t∈, which give rise; in part; to the following equations: satisfying for evolution parameters y, t∈ 2 the Lax-Sato vector field compatibility condition: As a simple consequence of the condition one finds exactly the first equation of the (85); coinciding with the heavenly type eqn. (80). Thereby; we have stated that this equation is a completely integrable heavenly type dynamical system with respect to both evolution parameters.

Remark 4.1:
It is worth to observe that the third equation of (85) entails the interesting relationship whose compatibility makes it possible to introduce a new function vC 2 ( 1 ;), satisfying the next differential expressions: which hold for all (x,y)∈ 1 ×. Based on (89) the seed element (81) is rewritten as and the vector fields (86) are rewritten as whose compatibility condition (87) gives rise to the following system of heavenly type nonliner integrable flows: compatible for arbitrary evolution parameters y,t∈.

The Hirota heavenly equation
The Hirota equation describes [41,53] The corresponding gradients for the Casimir invariants ( ) ( ), j I γ * ∈   = 1, 2, j are given by the following asymptotic expansions: as λ+α ≔µ→0, and as λ−α=→0. For the first case (96) one easily obtains that and for the second one (97) one obtains where we took into account that the following two Hamiltonian flows on *   It is easy now to check that the compatibility (108) for a set of the vector fields (109) gives rise to the Hirota heavenly eqn. (93); whose equivalent Lax-Sato vector field representation reads as a system of the linear vector field equations and λ∈C\{±α}.

The first reduced Shabat type heavenly equation
The entitled above equation [40] reads as where λ∈\{0,−1}. This element generates two independent hierarchies of Casimir functionals (1) (2) , ( ), I γ γ * ∈   whose gradient expansions are given by the following asymptotic expansions: one easily ensues from the compatibility condition for a set of the vector fields a compatible Lax-Sato representation as the following system of vector field equations:

General heavenly equation
This equation was first suggested and analyzed by Schief [21,53]; where it was shown to be equivalent to the first Pleba n′ski heavenly equation; and later studied by Doubrov and Ferapontov [54]; it has the form where α,β and γ∈ are arbitrary constants; satisfying the constraint α+β+=0, ( ; ), j j a b C ∞ ∈   = 0,1, j are smooth functions and µ∈ is a complex parameter. The corresponding equations for independent Casimir invariants ( ) ( ), = 1, 2, are given with respect to the standard metric ( , ) ⋅ ⋅ by the following asymptotic expansions: as µ+β=λ→0 and as µ−γ=→0. For the first case (124) one obtains that and for the second one (125) one finds that (2) 2 2 2 1 1 Here we took into account that the following two Hamiltonian flows on * with respect to the evolution parameters y, t∈ hold for the following conservation laws gradients: Owing to the compatibility condition of two commuting flows (129); one can easily rewrite it as the Lax relationship An easy calculation shows that the general heavenly eqn. (121) follows from the compatibility condition (130); whose equivalent vector field representation is given as for a function ( ; ),( , ) u C y t ∞ ∈ × ∈     and (x 1 ,x 2 )∈ 2 . To prove its Lax integrability; we define a seed element for a fixed function zC ∞ ( 2 ;). Then one easily obtains asymptotic expansionsas |λ|→∞ for coefficients of the two independent Casimir functionals (1) (2) , ( ) h h I * ∈   gradients: which can be rewritten as the compatibility condition for the following vector field equations:

Remark 4.2:
It is interesting to observe that the seed elements l * ∈    of the examples presented above have the differential geometric structure: where η and ω ω ω for any vector fields = ( ; ) , = ( ; )  on  1 and a fixed integer p∈.
TheThe integrable dynamical systems related to this central extension were described in detail [47]. Concerning a further generalization of the multi-dimensional case related to the loop group   for n + one can proceed in the following natural way: as the Lie algebra  �  ( ) n diff  consists of the elements formally depending additionally on the " spectral" variable λ∈ 1 , one can extend the basic Lie structure on diff( n ) to that on the adjacent holomorphic in It is now important to mention that the Lie algebra also splits into the direct sum of two subalgebras: allowing to introduce on it the classical -structure: for any , , a b ∈ where and := .
The space adjoint to the Lie algebra  of vector fields on × n , can be functionally identified with  subject to the Sobolev type metric are; respectively; defined for special integers p y ,p t ∈ + . These invariants generate; owing to the Lie-Poisson bracket (153); for the case q=0 the following commuting flows The latter allows to defiine on the Lie algebra   a new Lie bracket being equivalent to some system of nonlinear heavenly type equations in partial derivatives. Moreover; the system of evolution flows (194) The following examples demonstrate the analytical applicability of the devised above Lie-algebraic scheme for construction a wide class of nonlinear multidimensional heavenly type integrable Hamiltonian systems on functional spaces.
As is well known [17]; the invariant reduction of (200) at v=0 gives rise to the famous dispersionless Kadomtsev-Petviashvili equation

Example: The Dunajski heavenly equations
This equation; suggested [48]; generalizes the corresponding antiself-dual vacuum Einstein equation; which is related to the Pleban′ski metric and the celebrated Plebanski [16] second heavenly eqn. (116). To study the integrability of the Dunajski equations

∈
Taking also into account that all these Hamiltonian flows possess an infinite hierarchy of commuting nontrivial conservation laws; one can prove their formal complete integrability under some naturally formulated constraints. The corresponding analytical expressions for the infinite hierarchy of conservation laws can be retrieved from the asymptotic expansion (63) for Casimir functional gradients by employing the well-known [22,23,25,58] formal homotopy technique.
As an arbitrary heavenly equation is a Hamiltonian system with respect to both evolution parameters t,y∈ 2 and λ∈, one can construct [22,23,45,58] its suitable Lagrangian (or quasi-Lagrangian) representation under some natural constraints. Thus; it is possible to retrieve the corresponding Poisson structures related to both these evolution parameters t,y∈ 2 and λ∈, which; as follows from the Liealgebraic analysis in Section 3; are compatible to each other. In this way; one can show that any heavenly type equation is a bi-Hamiltonian integrable system on the corresponding functional manifold. It should be mentioned here that this property was introduced by Sergyeyev in (arXiv:1501.01955); published [53]; and rediscovered and applied in detail [30] for investigating the integrability properties of the general heavenly eqn. (121); first suggested by Schief [53] and later studied by Doubrov and Ferapontov [54].
In his book "Mecanique analytique"; v.1-2; published in 1788 in Paris; J.L. Lagrange formulated one of the basic; most general; differential variational principles of classical mechanics; expressing necessary and sufficient conditions for the correspondence of the real motion of a system of material points; subjected by ideal constraints; to the applied active forces. Within the d' Alembert-Lagrange principle the positions of the system in its real motion are compared with infinitely close positions permitted by the constraints at the given moment of time.
According to the d' Alembert-Lagrange principle; during a real motion of a system of N + particles with massess ( ) , = 1, , j F j N the sum of the elementary works performed by the given active forces As it was first demonstrated in the work [28]; in the last case of generalized reversible motions of a compressible elastic liquid; located in a one-connected open domain Ω t  n with the smooth boundary ∂ Ωt, t∈, in space  n ,n∈ + , the expression (240) can be rewritten as