APPENDIX
1.1
A1.
On Third-Order Resonances
We now list third-order resonances and display the values of \({{\omega }_{1}},{{\omega }_{2}},{{\mu }^{{(0)}}}\), and μ(2), which correspond to them.
$$\begin{gathered} 1)\;\;3{{\lambda }_{2}} = - 1,\quad {{\omega }_{1}} = \frac{{2\sqrt 2 }}{3},\quad {{\omega }_{2}} = \frac{1}{3}, \\ {{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {6177} }}{{162}} = 0.0148525130...,\quad {{\mu }^{{(2)}}} = - \frac{{304\sqrt {6177} }}{{277\,965}} = - 0.0859552225...; \\ \end{gathered} $$
(A1.1)
$$\begin{gathered} 2)\;\;{{\lambda }_{1}} + 2{{\lambda }_{2}} = 0,\quad {{\omega }_{1}} = \frac{{2\sqrt 5 }}{5},\quad {{\omega }_{2}} = \frac{{\sqrt 5 }}{5}, \\ {{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {1833} }}{{90}} = 0.0242938971...,\quad {{\mu }^{{(2)}}} = - \frac{{184\sqrt {1833} }}{{27\,495}} = - 0.2865136593...; \\ \end{gathered} $$
(A1.2)
$$\begin{gathered} 3)\;\;2{{\lambda }_{1}} + {{\lambda }_{2}} = 1,\quad {{\omega }_{1}} = \frac{4}{5},\quad {{\omega }_{2}} = \frac{3}{5}, \\ {{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {4857} }}{{150}} = 0.0353854644...,\quad {{\mu }^{{(2)}}} = \frac{{2336\sqrt {4857} }}{{1\,578\,525}} = 0.1031348463...; \\ \end{gathered} $$
(A1.3)
$$\begin{gathered} 4)\;\;{{\lambda }_{1}} - 2{{\lambda }_{2}} = 2,\quad {{\omega }_{1}} = \frac{4}{5},\quad {{\omega }_{2}} = \frac{3}{5}, \\ {{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {4857} }}{{150}} = 0.0353854644...,\quad {{\mu }^{{(2)}}} = \frac{{2272\sqrt {4857} }}{{526\,175}} = 0.3009277022...; \\ \end{gathered} $$
(A1.4)
$$\begin{gathered} 5)\;\;3{{\lambda }_{2}} = - 2,\quad {{\omega }_{1}} = \frac{{\sqrt 5 }}{3},\quad {{\omega }_{2}} = \frac{2}{3}, \\ {{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {5601} }}{{162}} = 0.0380257471...,\quad {{\mu }^{{(2)}}} = \frac{{520\sqrt {5601} }}{{352\,863}} = 0.1102884435.... \\ \end{gathered} $$
(A1.5)
Underlying the studied nonlinear problem is an analysis of the motions set by the normal form of the Hamiltonian function \(H({{q}_{1}},{{q}_{2}},{{q}_{3}},{{p}_{1}},{{p}_{2}},{{p}_{3}},\nu ;\mu ,e)\). The latter can be found using a Brinkhoff transformation close to the identical \({{y}_{j}},{{Y}_{j}} \to {{q}_{j}},{{p}_{j}}(j = 1,2,3)\), which is \(2\pi \)-periodic in \(\nu \) and analytical in e and the newly introduced canonically conjugated variables \({{q}_{j}},{{p}_{j}}.\)
It should be noted that the variable \({{{v}}_{3}}\) in the initial function (1.2) is not contained in the \({{\Gamma }_{m}}\) form of the power higher than two, while the variable u3 is only present in even powers. We also take into account that the values \({{\lambda }_{j}}\) \((j = 1,2,)\) from Eqs. (2.3) and (2.4) inside the stability area in the first approximation fulfill the inequality \({\text{0}} < \left| {{{\lambda }_{j}}} \right| < {\text{1}}\). Taking this observation into account, it can be shown, following [8], that the presence of identity resonance \({{\lambda }_{{\text{3}}}} = 1\) of the spatial problem does not affect the structure of the normal form of the terms of the third power with respect to the new variables \({{q}_{j}},{{p}_{j}}(j = 1,2,3).\) They will be the same as in the case of a planar elliptic problem.
Setting \({{q}_{j}} = \sqrt {2{{r}_{j}}} \sin {{\varphi }_{j}}\), \({{p}_{j}} = \sqrt {2{{r}_{j}}} \cos {{\varphi }_{j}}\) \((j = 1,2,3)\), we can represent the Hamiltonian function normalized up to terms of the fourth power in \({{q}_{j}},{{p}_{j}}(j = 1,2,3)\), in the form
$$H = {{\lambda }_{1}}{{r}_{1}} + {{\lambda }_{2}}{{r}_{2}} + {{r}_{3}} + {{H}_{3}} + {{H}_{4}} + O\left( {{{{({{r}_{1}} + {{r}_{2}} + {{r}_{3}})}}^{{5{\text{/}}2}}}} \right).$$
(A1.6)
The fourth-power terms \({{H}_{4}}\) can be found using a formula similar to Eq. (3.4) in Chapter 10 of book [8]. We represent the structure of the third-power terms \({{H}_{3}}\) for the third-order resonances (2.15).
$$1)\;\;3{{\lambda }_{2}} = - 1,\quad {{H}_{3}} = r_{2}^{{3/2}}[f_{{0,3}}^{{( - 1)}}\sin (3{{\varphi }_{2}} + \nu ) + g_{{0,3}}^{{( - 1)}}\cos (3{{\varphi }_{2}} + \nu )];$$
$$2)\;\;{{\lambda }_{1}} + 2{{\lambda }_{2}} = 0,\quad {{H}_{3}} = r_{1}^{{1/2}}{{r}_{2}}[f_{{1,2}}^{{(0)}}\sin ({{\varphi }_{1}} + 2{{\varphi }_{2}}) + g_{{1,2}}^{{(0)}}\cos ({{\varphi }_{1}} + 2{{\varphi }_{2}})];$$
$$3)\;\;2{{\lambda }_{1}} + {{\lambda }_{2}} = 1,\quad {{H}_{3}} = {{r}_{1}}r_{2}^{{1/2}}[f_{{2,1}}^{{(1)}}\sin (2{{\varphi }_{1}} + {{\varphi }_{2}} - \nu ) + g_{{2,1}}^{{(1)}}\cos (2{{\varphi }_{1}} + {{\varphi }_{2}} - \nu )];$$
$$4)\;\;{{\lambda }_{1}} - 2{{\lambda }_{2}} = 2,\quad {{H}_{3}} = r_{1}^{{1/2}}{{r}_{2}}[f_{{1, - 2}}^{{(2)}}\sin ({{\varphi }_{1}} - 2{{\varphi }_{2}} - 2\nu ) + g_{{1, - 2}}^{{(2)}}\cos ({{\varphi }_{1}} - 2{{\varphi }_{2}} - 2\nu )];$$
$$5)\;\;3{{\lambda }_{2}} = - 2,\quad {{H}_{3}} = r_{2}^{{3/2}}[f_{{0,3}}^{{( - 2)}}\sin (3{{\varphi }_{2}} + 2\nu ) + g_{{0,3}}^{{( - 2)}}\cos (3{{\varphi }_{2}} + 2\nu )].$$
In the formulas for \({{H}_{3}}\), quantities \(f_{{{{s}_{1}},{{s}_{2}}}}^{{(n)}}\) and \(g_{{{{s}_{1}},{{s}_{2}}}}^{{(n)}}\) are functions of \(\mu \) and e of the order of \({{e}^{{\left| n \right|}}}\).
1.2
A2.
Fourth-order resonances
$$(1)\;\;4{{\lambda }_{2}} = - 1,\quad {{\omega }_{1}} = \frac{{\sqrt {15} }}{4},\quad {{\omega }_{2}} = \frac{1}{4},$$
(A2.1)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {139} }}{{24}} = 0.0087572448...,\quad {{\mu }^{{(2)}}} = - \frac{{265\sqrt {139} }}{{80\,064}} = - 0.0390225809...,$$
$$(2)\;\;{{\lambda }_{1}} + 3{{\lambda }_{2}} = 0,\quad {{\omega }_{1}} = \frac{{3\sqrt {10} }}{{10}},\quad {{\omega }_{2}} = \frac{{\sqrt {10} }}{{10}}$$
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {213} }}{{30}} = 0.0135160160...,\quad {{\mu }^{{(2)}}} = - \frac{{62\sqrt {213} }}{{13\,845}} = - 0.0653564615...,$$
$$(3)\;\;{{\lambda }_{1}} - 3{{\lambda }_{2}} = 2,\quad {{\omega }_{1}} = \frac{1}{5} + \frac{{3\sqrt 6 }}{{10}},\quad {{\omega }_{2}} = \frac{3}{5} - \frac{{\sqrt 6 }}{{10}}$$
(A2.3)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {5101 + 64\sqrt 6 } }}{{150}} = 0.0165969087...,$$
$${{\mu }^{{(2)}}} = \frac{{2(8456 - 12\,841\sqrt 6 )\sqrt {41\,593(5101 - 64\sqrt 6 )} }}{{5\,381\,094\,375}} = - 0.12257646673...;$$
$$(4)\;\;2{{\lambda }_{1}} + 2{{\lambda }_{2}} = 1,\quad {{\omega }_{1}} = \frac{1}{4} + \frac{{\sqrt 7 }}{4},\quad {{\omega }_{2}} = - \frac{1}{4} + \frac{{\sqrt 7 }}{4},$$
(A2.4)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {33} }}{{12}} = 0.0212864461...,\quad {{\mu }^{{(2)}}} = - \frac{{25\sqrt {33} }}{{1056}} = - 0.1359981687...;$$
$$(5)\;\;3{{\lambda }_{1}} + {{\lambda }_{2}} = 2,\quad {{\omega }_{1}} = \frac{3}{5} + \frac{{\sqrt 6 }}{{10}},\quad {{\omega }_{2}} = - \frac{1}{5} + \frac{{3\sqrt 6 }}{{10}},$$
(A2.5)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {5101 - 64\sqrt 6 } }}{{150}} = 0.0312317484...,$$
$${{\mu }^{{(2)}}} = \frac{{2(8456 + 12\,841\sqrt 6 )\sqrt {41\,593(5101 + 64\sqrt 6 )} }}{{5\,381\,094\,375}} = 0.2193566513...;$$
$$(6)\;\;3{{\lambda }_{1}} - {{\lambda }_{2}} = 3,\quad {{\omega }_{1}} = \frac{4}{5},\quad {{\omega }_{2}} = \frac{3}{5},$$
(A2.6)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {4857} }}{{150}} = 0.0353854644...,\quad {{\mu }^{{(2)}}} = - \frac{{2144\sqrt {4857} }}{{1\,578\,525}} = - 0.0946580096...;$$
$$(7)\;\;{{\lambda }_{1}} + 3{{\lambda }_{2}} = - 1,\quad {{\omega }_{1}} = \frac{4}{5},\quad {{\omega }_{2}} = \frac{3}{5},$$
(A2.7)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {4857} }}{{150}} = 0.0353854644...,\quad {{\mu }^{{(2)}}} = \frac{{11\,488\sqrt {4857} }}{{4\,735\,575}} = 0.1690657982...;$$
$$(8)\;\;4{{\lambda }_{1}} = 3,\quad {{\omega }_{1}} = \frac{3}{4},\quad {{\omega }_{2}} = \frac{{\sqrt 7 }}{4},$$
(A2.8)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {123} }}{{24}} = 0.0378943122...,\quad {{\mu }^{{(2)}}} = \frac{{203\sqrt {123} }}{{39\,360}} = 0.0571996674....$$
1.3
A3.
Fifth-order resonances
$$(1)\;\;5{{\lambda }_{2}} = - 1,\quad {{\omega }_{1}} = \frac{{2\sqrt 6 }}{5},\quad {{\omega }_{2}} = \frac{1}{5}$$
(A3.1)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {5497} }}{{150}} = 0.0057216258...,\quad {{\mu }^{{(2)}}} = - \frac{{2704\sqrt {5497} }}{{8\,657\,775}} = - 0.0231559850...$$
$$(2)\;\;{{\lambda }_{1}} + 4{{\lambda }_{2}} = 0,\quad {{\omega }_{1}} = \frac{{4\sqrt {17} }}{{17}},\quad {{\omega }_{2}} = \frac{{\sqrt {17} }}{{17}};$$
(A3.2)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {22\,641} }}{{306}} = 0.0082703726...,\quad {{\mu }^{{(2)}}} = - \frac{{9376\sqrt {22\,641} }}{{41\,500\,953}} = - 0.0339943961...$$
$$(3)\;\;{{\lambda }_{1}} - 4{{\lambda }_{2}} = 2,\quad {{\omega }_{1}} = \frac{2}{{17}} + \frac{{4\sqrt {13} }}{{17}},\quad {{\omega }_{2}} = \frac{8}{{17}} - \frac{{\sqrt {13} }}{{17}}$$
(A3.3)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {682\,377 + 11\,520\sqrt {13} } }}{{1734}} = 0.0093248318...,$$
$${{\mu }^{{(2)}}} = \frac{{40(2\,262\,155 - 2\,045\,476\sqrt {13} )\sqrt {617\,161(682\,377 - 11\,520\sqrt {13} )} }}{{2\,820\,437\,222\,656\,677}} = - 0.0456022641...;$$
$$(4)\;\;2{{\lambda }_{1}} + 3{{\lambda }_{2}} = 1,\quad {{\omega }_{1}} = \frac{2}{{13}} + \frac{{6\sqrt 3 }}{{13}},\quad {{\omega }_{2}} = - \frac{3}{{13}} + \frac{{4\sqrt 3 }}{{13}},$$
(A3.4)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {232\,217 + 7040\sqrt 3 } }}{{1014}} = 0.0124467036...,$$
$${{\mu }^{{(2)}}} = \frac{{80(420 - 1093\sqrt 3 )\sqrt {1\,882\,849(232\,217 - 7040\sqrt 3 )} }}{{1\,451\,953\,357\,803}} = - 0.0522421439...;$$
$$(5)\;\;2{{\lambda }_{1}} - 3{{\lambda }_{2}} = 3,\quad {{\omega }_{1}} = \frac{{12}}{{13}},\quad {{\omega }_{2}} = \frac{5}{{13}},$$
(A3.5)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {237\,849} }}{{1014}} = 0.0190358459...,\quad {{\mu }^{{(2)}}} = - \frac{{14\,317\,600\sqrt {237\,849} }}{{34\,207\,205\,331}} = - 0.2041283360...;$$
$$(6)\;\;3{{\lambda }_{1}} + 2{{\lambda }_{2}} = 2,\quad {{\omega }_{1}} = \frac{{12}}{{13}},\quad {{\omega }_{2}} = \frac{5}{{13}},$$
(A3.6)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {237\,849} }}{{1014}} = 0.0190358459...,$$
$${{\mu }^{{(2)}}} = - \frac{{18\,200\,800\sqrt {237\,849} }}{{102\,621\,615\,993}} = - 0.0864972485...,$$
$$(7)\;\;5{{\lambda }_{2}} = - 2,\quad {{\omega }_{1}} = \frac{{\sqrt {21} }}{5},\quad {{\omega }_{2}} = \frac{2}{5},$$
(A3.7)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {5177} }}{{150}} = 0.0203241835...,\quad {{\mu }^{{(2)}}} = - \frac{{8456\sqrt {5177} }}{{3\,494\,475}} = - 0.1741093599...;$$
$$(8)\;\;{{\lambda }_{1}} + 4{{\lambda }_{2}} = - 1,\quad {{\omega }_{1}} = \frac{{15}}{{17}},\quad {{\omega }_{2}} = \frac{8}{{17}},$$
(A3.8)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {674\,889} }}{{1734}} = 0.0262305184...,$$
$${{\mu }^{{(2)}}} = - \frac{{28\,476\,800\sqrt {674\,889} }}{{39\,723\,741\,577}} = - 0.5889212320...;$$
$$(9)\;\;4{{\lambda }_{1}} - {{\lambda }_{2}} = 4,\quad {{\omega }_{1}} = \frac{{15}}{{17}},\quad {{\omega }_{2}} = \frac{8}{{17}},$$
(A3.9)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {674\,889} }}{{1734}} = 0.0262305184...,$$
$${{\mu }^{{(2)}}} = \frac{{89\,737\,600\sqrt {674\,889} }}{{119\,171\,224\,731}} = 0.6186132565...;$$
$$(10)\;\;4{{\lambda }_{1}} + {{\lambda }_{2}} = 3,\quad {{\omega }_{1}} = \frac{{12}}{{17}} + \frac{{2\sqrt 2 }}{{17}},\quad {{\omega }_{2}} = - \frac{3}{{17}} + \frac{{8\sqrt 2 }}{{17}},$$
(A3.10)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {5\,986\,833 + 34\,560\sqrt 2 } }}{{5202}} = 0.0277262898...,$$
$${{\mu }^{{(2)}}} = \frac{{80(1\,738\,235 - 24\,462\,756\sqrt 2 )\sqrt {47\,679\,001(5\,986\,833 - 34\,560\sqrt 2 )} }}{{79\,958\,550\,479\,979\,159}} = - 0.5531455483...;$$
$$(11)\;\;{{\lambda }_{1}} - 4{{\lambda }_{2}} = 3,\quad {{\omega }_{1}} = \frac{3}{{17}} + \frac{{8\sqrt 2 }}{{17}},\quad {{\omega }_{2}} = \frac{{12}}{{17}} - \frac{{2\sqrt 2 }}{{17}},$$
(A3.11)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {5\,986\,833 - 34\,560\sqrt 2 } }}{{5202}} = 0.0315662209...,$$
$${{\mu }^{{(2)}}} = \frac{{80(1\,738\,235 + 24\,462\,756\sqrt 2 )\sqrt {47\,679\,001(5\,986\,833 + 34\,560\sqrt 2 )} }}{{79\,958\,550\,479\,979\,159}} = 0.6166855225...;$$
$$(12)\;\;2{{\lambda }_{1}} + 3{{\lambda }_{2}} = 0,\quad {{\omega }_{1}} = \frac{{3\sqrt {13} }}{{13}},\quad {{\omega }_{2}} = \frac{{2\sqrt {13} }}{{13}},$$
(A3.12)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {1329} }}{{78}} = 0.0326224068...,\quad {{\mu }^{{(2)}}} = \frac{{2936\sqrt {1329} }}{{397\,371}} = 0.2693533444...;$$
$$(13)\;\;5{{\lambda }_{1}} = 4,\quad {{\omega }_{1}} = \frac{4}{5},\quad {{\omega }_{2}} = \frac{3}{5},$$
(A3.13)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {4857} }}{{150}} = 0.0353854644...,\quad {{\mu }^{{(2)}}} = \frac{{2528\sqrt {4857} }}{{4\,735\,575}} = 0.0372038943...;$$
$$(14)\;\;5{{\lambda }_{2}} = - 3,\quad {{\omega }_{1}} = \frac{4}{5},\quad {{\omega }_{2}} = \frac{3}{5},$$
(A3.14)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {4857} }}{{150}} = 0.0353854644...,\quad {{\mu }^{{(2)}}} = \frac{{352\sqrt {4857} }}{{121\,425}} = 0.2020312742...;$$
$$(15)\;\;3{{\lambda }_{1}} + 2{{\lambda }_{2}} = 1,\quad {{\omega }_{1}} = \frac{3}{{13}} + \frac{{4\sqrt 3 }}{{13}},\quad {{\omega }_{2}} = - \frac{2}{{13}} + \frac{{6\sqrt 3 }}{{13}},$$
(A3.15)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {232\,217 - 7040\sqrt 3 } }}{{1014}} = 0.0374097834...,$$
$${{\mu }^{{(2)}}} = \frac{{80(420 + 1093\sqrt 3 )\sqrt {1\,882\,849(232\,217 + 7040\sqrt 3 )} }}{{1\,451\,953\,357\,803}} = 0.0864580524...;$$
$$(16)\;\;{{\lambda }_{1}} + 4{{\lambda }_{2}} = - 2,\quad {{\omega }_{1}} = - \frac{2}{{17}} + \frac{{4\sqrt {13} }}{{17}},\quad {{\omega }_{2}} = \frac{8}{{17}} + \frac{{\sqrt {13} }}{{17}},$$
(A3.16)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {682\,377 - 11\,520\sqrt {13} } }}{{1734}} = 0.0383359381...,$$
$${{\mu }^{{(2)}}} = \frac{{40(2\,262\,155 + 2\,045\,476\sqrt {13} )\sqrt {617\,161(682\,377 + 11\,520\sqrt {13} )} }}{{2\,820\,437\,222\,656\,677}} = 0.0913561732....;$$
1.4
A4.
Sixth-order resonances
$$(1)\;\;6{{\lambda }_{2}} = - 1,\quad {{\omega }_{1}} = \frac{{\sqrt {35} }}{6},\quad {{\omega }_{2}} = \frac{1}{6},$$
(A4.1)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {1614} }}{{81}} = 0.0040170511...,\quad {{\mu }^{{(2)}}} = - \frac{{1435\sqrt {1614} }}{{3\,718\,656}} = - 0.0155030683...;$$
$$(2)\;\;{{\lambda }_{1}} + 5{{\lambda }_{2}} = 0,\quad {{\omega }_{1}} = \frac{{5\sqrt {26} }}{{26}},\quad {{\omega }_{2}} = \frac{{\sqrt {26} }}{{26}},$$
(A4.2)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {13\,389} }}{{234}} = 0.0055092029...,\quad {{\mu }^{{(2)}}} = - \frac{{3550\sqrt {13\,389} }}{{19\,320\,327}} = - 0.0212612087...;$$
$$(3)\;\;{{\lambda }_{1}} - 5{{\lambda }_{2}} = 2,\quad {{\omega }_{1}} = \frac{1}{{13}} + \frac{{5\sqrt {22} }}{{26}},\quad {{\omega }_{2}} = \frac{5}{{13}} - \frac{{\sqrt {22} }}{{26}},$$
(A4.3)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {237\,453 + 2880\sqrt {22} } }}{{1014}} = 0.0059561391...,$$
$${{\mu }^{{(2)}}} = \frac{{2(25\,504 - 6583\sqrt {22} )\sqrt {218\,641(237\,453 - 2880\sqrt {22} )} }}{{93\,669\,084\,015}} = - 0.0253855879...;$$
$$(4)\;\;2{{\lambda }_{1}} + 4{{\lambda }_{2}} = 1,\quad {{\omega }_{1}} = \frac{1}{{10}} + \frac{{\sqrt {19} }}{5},\quad {{\omega }_{2}} = - \frac{1}{5} + \frac{{\sqrt {19} }}{{10}},$$
(A4.4)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {1284 + 18\sqrt {19} } }}{{75}} = 0.0078464230...,$$
$${{\mu }^{{(2)}}} = \frac{{(175\,427 - 71\,171\sqrt {19} )\sqrt {73(1284 - 18\sqrt {19} )} }}{{1\,331\,520\,000}} = - 0.0300327459...;$$
$$(5)\;\;2{{\lambda }_{1}} - 4{{\lambda }_{2}} = 3,\quad {{\omega }_{1}} = \frac{3}{{10}} + \frac{{\sqrt {11} }}{5},\quad {{\omega }_{2}} = \frac{3}{5} - \frac{{\sqrt {11} }}{{10}},$$
(A4.5)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {11\,976 + 54\sqrt {11} } }}{{225}} = 0.0099992897...,$$
$${{\mu }^{{(2)}}} = \frac{{(2133\sqrt {11} - 122\,573)\sqrt {6373(11\,976 - 54\sqrt {11} )} }}{{18\,354\,240\,000}} = - 0.0545627701...;$$
$$(6)\;\;3{{\lambda }_{1}} + 3{{\lambda }_{2}} = 2,\quad {{\omega }_{1}} = \frac{1}{3} + \frac{{\sqrt {14} }}{6},\quad {{\omega }_{2}} = - \frac{1}{3} + \frac{{\sqrt {14} }}{6},$$
(A4.6)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {6261} }}{{162}} = 0.0115649319...,\quad {{\mu }^{{(2)}}} = - \frac{{400\sqrt {6261} }}{{732\,537}} = - 0.0432068174...;$$
$$(7)\;\;4{{\lambda }_{1}} + 2{{\lambda }_{2}} = 3,\quad {{\omega }_{1}} = \frac{3}{5} + \frac{{\sqrt {11} }}{{10}},\quad {{\omega }_{2}} = - \frac{3}{{10}} + \frac{{\sqrt {11} }}{5},$$
(A4.7)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {11\,976 - 54\sqrt {11} } }}{{225}} = 0.0172731312...,$$
$${{\mu }^{{(2)}}} = - \frac{{(122\,573 + 2133\sqrt {11} )\sqrt {6373(11\,976 + 54\sqrt {11} )} }}{{18\,354\,240\,000}} = - 0.06216965547...;$$
$$(8)\;\;{{\lambda }_{1}} + 5{{\lambda }_{2}} = - 1,\quad {{\omega }_{1}} = \frac{{12}}{{13}},\quad {{\omega }_{2}} = \frac{5}{{13}},$$
(A4.8)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {237\,849} }}{{1014}} = 0.0190358459...,$$
$${{\mu }^{{(2)}}} = - \frac{{3\,122\,400\sqrt {237\,849} }}{{11\,402\,401\,777}} = - 0.1335496835...;$$
$$(9)\;\;5{{\lambda }_{1}} - {{\lambda }_{2}} = 5,\quad {{\omega }_{1}} = \frac{{12}}{{13}},\quad {{\omega }_{2}} = \frac{5}{{13}},$$
(A4.9)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {237\,849} }}{{1014}} = 0.0190358459...,\quad {{\mu }^{{(2)}}} = \frac{{10\,434\,400\sqrt {237\,849} }}{{34\,207\,205\,331}} = 0.1487649263...;$$
$$(10)\;\;{{\lambda }_{1}} - 5{{\lambda }_{2}} = 3,\quad {{\omega }_{1}} = \frac{3}{{26}} + \frac{{5\sqrt {17} }}{{26}},\quad {{\omega }_{2}} = \frac{{15}}{{26}} - \frac{{\sqrt {17} }}{{26}},$$
(A4.10)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {2\,044\,257 + 17\,280\sqrt {17} } }}{{3042}} = 0.0218680746...,$$
$${{\mu }^{{(2)}}} = - \frac{{16(20\,451\,221 + 8\,794\,683\sqrt {17} )\sqrt {16\,237\,801(2\,044\,257 - 17\,280\sqrt {17} )} }}{{20\,932\,161\,203\,883\,735}} = - 0.2453657301...;$$
$$(11)\;\;3{{\lambda }_{1}} - 3{{\lambda }_{2}} = 4,\quad {{\omega }_{1}} = \frac{2}{3} + \frac{{\sqrt 2 }}{6},\quad {{\omega }_{2}} = \frac{2}{3} - \frac{{\sqrt 2 }}{6},$$
(A4.11)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {5973} }}{{162}} = 0.0229309496...,\quad {{\mu }^{{(2)}}} = - \frac{{17\,150\sqrt {5973} }}{{2\,526\,579}} = - 0.5245990498...;$$
$$(12)\;\;5{{\lambda }_{1}} + {{\lambda }_{2}} = 4,\quad {{\omega }_{1}} = \frac{{10}}{{13}} + \frac{{\sqrt {10} }}{{26}},\quad {{\omega }_{2}} = - \frac{2}{{13}} + \frac{{5\sqrt {10} }}{{26}},$$
(A4.12)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {226\,029 + 1920\sqrt {10} } }}{{1014}} = 0.0248834579...,$$
$${{\mu }^{{(2)}}} = \frac{{10(10\,703 - 26\,746\sqrt {10} )\sqrt {198\,609(226\,029 - 1920\sqrt {10} )} }}{{1\,412\,445\,440\,601}} = - 0.1093191450...;$$
$$(13)\;\;3{{\lambda }_{1}} + 3{{\lambda }_{2}} = 1,\quad {{\omega }_{1}} = \frac{1}{6} + \frac{{\sqrt {17} }}{6},\quad {{\omega }_{2}} = - \frac{1}{6} + \frac{{17}}{6},$$
(A4.13)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {5793} }}{{162}} = 0.0301743214...,\quad {{\mu }^{{(2)}}} = \frac{{5600\sqrt {5793} }}{{677\,781}} = 0.6288548301...;$$
$$(14)\;\;6{{\lambda }_{1}} = 5,\quad {{\omega }_{1}} = \frac{5}{6},\quad {{\omega }_{2}} = \frac{{\sqrt {11} }}{6},$$
(A4.14)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {1434} }}{{81}} = 0.0324914511...,\quad {{\mu }^{{(2)}}} = \frac{{4675\sqrt {1434} }}{{6\,607\,872}} = 0.0267913482...;$$
$$(15)\;\;{{\lambda }_{1}} + 5{{\lambda }_{2}} = - 2,\quad {{\omega }_{1}} = - \frac{1}{{13}} + \frac{{5\sqrt {22} }}{{26}},\quad {{\omega }_{2}} = \frac{5}{{13}} + \frac{{\sqrt {22} }}{{26}},$$
(A4.15)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {237\,453 - 2880\sqrt {22} } }}{{1014}} = 0.0333058630...,$$
$${{\mu }^{{(2)}}} = \frac{{2(25\,504 + 6583\sqrt {22} )\sqrt {218\,641(237\,453 + 2880\sqrt {22} )} }}{{93\,669\,084\,015}} = 0.2819913682...;$$
$$(16)\;\;2{{\lambda }_{1}} + 4{{\lambda }_{2}} = - 1,\quad {{\omega }_{1}} = - \frac{1}{{10}} + \frac{{\sqrt {19} }}{5},\quad {{\omega }_{2}} = \frac{1}{5} + \frac{{\sqrt {19} }}{{10}},$$
(A4.16)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {1284 - 18\sqrt {19} } }}{{75}} = 0.0370548736...,$$
$${{\mu }^{{(2)}}} = \frac{{(175\,427 + 71\,171\sqrt {19} )\sqrt {73(1284 - 18\sqrt {19} )} }}{{1\,331\,520\,000}} = 0.1150277940...;$$
$$(17)\;\;{{\lambda }_{1}} - 5{{\lambda }_{2}} = 4,\quad {{\omega }_{1}} = \frac{2}{{13}} + \frac{{5\sqrt {10} }}{{26}},\quad {{\omega }_{2}} = \frac{{10}}{{13}} - \frac{{\sqrt {10} }}{{26}},$$
(A4.17)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {226\,029 - 1920\sqrt {10} } }}{{1014}} = 0.0374791018...,$$
$${{\mu }^{{(2)}}} = \frac{{10(10\,703 + 26\,746\sqrt {10} )\sqrt {198\,609(226\,029 + 1920\sqrt {10} )} }}{{1\,412\,445\,440\,601}} = 0.1448349796...;$$
$$(18)\;\;5{{\lambda }_{1}} + {{\lambda }_{2}} = 3,\quad {{\omega }_{1}} = \frac{{15}}{{26}} + \frac{{\sqrt {17} }}{{26}},\quad {{\omega }_{2}} = - \frac{3}{{26}} + \frac{{5\sqrt {17} }}{{26}},$$
(A4.18)
$${{\mu }^{{(0)}}} = \frac{1}{2} - \frac{{\sqrt {2\,044\,257 - 17\,280\sqrt {17} } }}{{3042}} = 0.0382515946...,$$
$${{\mu }^{{(2)}}} = \frac{{16( - 20\,451\,221 + 8\,794\,683\sqrt {17} )\sqrt {16\,237\,801(2\,044\,257 + 17\,280\sqrt {17} )} }}{{20\,932\,161\,203\,883\,735}} = 0.0708293815....;$$
This work was supported by a grant from the Russian Science Foundation (project no. 19-11-00116) at the Moscow Aviation Institute (National Research University)
