doi: 10.3934/dcdsb.2021228
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Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Catalonia, Spain

2. 

School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082, China

* Corresponding author: Yuzhou Tian

Received  November 2020 Revised  August 2021 Early access September 2021

We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian $ H = \left(p_1^2+p_2^2\right)/2+1/P(q_1, q_2) $, being $ P(q_1, q_2) $ a homogeneous polynomial of degree $ 4 $ of one of the following forms $ \pm q_1^4 $, $ 4q_1^3q_2 $, $ \pm 6q_1^2q_2^2 $, $ \pm \left(q_1^2+q_2^2\right)^2 $, $ \pm q_2^2\left(6q_1^2-q_2^2\right) $, $ \pm q_2^2\left(6q_1^2+q_2^2\right) $, $ q_1^4+6\mu q_1^2q_2^2-q_2^4 $, $ -q_1^4+6\mu q_1^2q_2^2+q_2^4 $ with $ \mu>-1/3 $ and $ \mu\neq 1/3 $, and $ q_1^4+6\mu q_1^2q_2^2+q_2^4 $ with $ \mu \neq \pm 1/3 $. We note that any homogeneous polynomial of degree $ 4 $ after a linear change of variables and a rescaling can be written as one of the previous polynomials. We remark that for the polynomial $ q_1^4+6\mu q_1^2q_2^2+q_2^4 $ when $ \mu\in\left\{-5/3, -2/3\right\} $ we only can prove that it has no a polynomial first integral.

Citation: Jaume Llibre, Yuzhou Tian. Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021228
References:
[1]

T. BountisH. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painlevé property, Phys. Rev. A (3), 25 (1982), 1257-1264.  doi: 10.1103/PhysRevA.25.1257.  Google Scholar

[2]

Y. F. ChangM. Tabor and J. Weiss, Analytic structure of the Hénon-Heiles Hamiltonian in integrable and nonintegrable regimes, J. Math. Phys., 23 (1982), 531-538.  doi: 10.1063/1.525389.  Google Scholar

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G. Duval and A. J. Maciejewski, Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials, Ann. Inst. Fourier (Grenoble), 59 (2009), 2839-2890.  doi: 10.5802/aif.2510.  Google Scholar

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G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations, Discrete Contin. Dyn. Syst., 34 (2014), 4589-4615.  doi: 10.3934/dcds.2014.34.4589.  Google Scholar

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G. Duval and A. J. Maciejewski, Integrability of potentials of degree $k\neq\pm 2$. Second order variational equations between Kolchin solvability and Abelianity, Discrete Contin. Dyn. Syst., 35 (2015), 1969-2009.  doi: 10.3934/dcds.2015.35.1969.  Google Scholar

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B. GrammaticosB. Dorizzi and A. Ramani, Integrability of Hamiltonians with third- and fourth-degree polynomial potentials, J. Math. Phys., 24 (1983), 2289-2295.  doi: 10.1063/1.525976.  Google Scholar

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L. S. Hall, A theory of exact and approximate configurational invariants, Phys. D, 8 (1983), 90-116.  doi: 10.1016/0167-2789(83)90312-3.  Google Scholar

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J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rep., 147 (1987), 87-154.  doi: 10.1016/0370-1573(87)90089-5.  Google Scholar

[11]

J. LlibreA. Mahdi and C. Valls, Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree $-3$, Phys. D, 240 (2011), 1928-1935.  doi: 10.1016/j.physd.2011.09.003.  Google Scholar

[12]

J. Llibre, A. Mahdi and C. Valls, Analytic integrability of Hamiltonian systems with a homogeneous polynomial potential of degree 4, J. Math. Phys., 52 (2011), 012702, 9 pp. doi: 10.1063/1.3544473.  Google Scholar

[13]

J. LlibreA. Mahdi and C. Valls, Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree $-2$, Phys. Lett. A, 375 (2011), 1845-1849.  doi: 10.1016/j.physleta.2011.03.042.  Google Scholar

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J. Llibre and C. Valls, Darboux integrability of 2-dimensional Hamiltonian systems with homogenous potentials of degree 3, J. Math. Phys., 55 (2014), 033507, 12 pp. doi: 10.1063/1.4868701.  Google Scholar

[15]

J. Llibre and C. Valls, On the integrability of Hamiltonian systems with $d$ degrees of freedom and homogenous polynomial potential of degree $n$, Commun. Contemp. Math., 20 (2018), 1750045, 9 pp. doi: 10.1142/S0219199717500456.  Google Scholar

[16]

A. J. Maciejewski and M. Przybylska, All meromorphically integrable 2D Hamiltonian systems with homogeneous potential of degree 3, Phys. Lett. A, 327 (2004), 461-473.  doi: 10.1016/j.physleta.2004.05.042.  Google Scholar

[17]

A. J. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential, J. Math. Phys., 46 (2005), 062901, 33 pp. doi: 10.1063/1.1917311.  Google Scholar

[18]

J. J. Morales Ruiz, Differential Galois theory and non-integrability of Hamiltonian systems, vol. 179 of Progress in Mathematics, Birkhäuser Verlag, Basel, 1999. doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[19]

A. RamaniB. Dorizzi and B. Grammaticos, Painlevé conjecture revisited, Phys. Rev. Lett., 49 (1982), 1539-1541.  doi: 10.1103/PhysRevLett.49.1539.  Google Scholar

[20]

H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Phys. D, 29 (1987), 128-142.  doi: 10.1016/0167-2789(87)90050-9.  Google Scholar

[21]

H. Yoshida, A new necessary condition for the integrability of Hamiltonian systems with a two-dimensional homogeneous potential, Phys. D, 128 (1999), 53-69.  doi: 10.1016/S0167-2789(98)00313-3.  Google Scholar

[22]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, Developments in Mathematics vol. 47, Springer Natural, Singapore, 2017. doi: 10.1007/978-981-10-4226-3.  Google Scholar

[23]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I, Funktsional. Anal. i Prilozhen., 16 (1982), 30–41, 96.  Google Scholar

[24]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II, Funktsional. Anal. i Prilozhen., 17 (1983), 8-23.   Google Scholar

show all references

References:
[1]

T. BountisH. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painlevé property, Phys. Rev. A (3), 25 (1982), 1257-1264.  doi: 10.1103/PhysRevA.25.1257.  Google Scholar

[2]

Y. F. ChangM. Tabor and J. Weiss, Analytic structure of the Hénon-Heiles Hamiltonian in integrable and nonintegrable regimes, J. Math. Phys., 23 (1982), 531-538.  doi: 10.1063/1.525389.  Google Scholar

[3]

A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic vector fields in the plane, J. Math. Anal. Appl., 147 (1990), 420-448.  doi: 10.1016/0022-247X(90)90359-N.  Google Scholar

[4]

G. Duval and A. J. Maciejewski, Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials, Ann. Inst. Fourier (Grenoble), 59 (2009), 2839-2890.  doi: 10.5802/aif.2510.  Google Scholar

[5]

G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations, Discrete Contin. Dyn. Syst., 34 (2014), 4589-4615.  doi: 10.3934/dcds.2014.34.4589.  Google Scholar

[6]

G. Duval and A. J. Maciejewski, Integrability of potentials of degree $k\neq\pm 2$. Second order variational equations between Kolchin solvability and Abelianity, Discrete Contin. Dyn. Syst., 35 (2015), 1969-2009.  doi: 10.3934/dcds.2015.35.1969.  Google Scholar

[7]

A. Goriely, Integrability and Nonintegrability of Dynamical Systems, vol. 19 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812811943.  Google Scholar

[8]

B. GrammaticosB. Dorizzi and A. Ramani, Integrability of Hamiltonians with third- and fourth-degree polynomial potentials, J. Math. Phys., 24 (1983), 2289-2295.  doi: 10.1063/1.525976.  Google Scholar

[9]

L. S. Hall, A theory of exact and approximate configurational invariants, Phys. D, 8 (1983), 90-116.  doi: 10.1016/0167-2789(83)90312-3.  Google Scholar

[10]

J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rep., 147 (1987), 87-154.  doi: 10.1016/0370-1573(87)90089-5.  Google Scholar

[11]

J. LlibreA. Mahdi and C. Valls, Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree $-3$, Phys. D, 240 (2011), 1928-1935.  doi: 10.1016/j.physd.2011.09.003.  Google Scholar

[12]

J. Llibre, A. Mahdi and C. Valls, Analytic integrability of Hamiltonian systems with a homogeneous polynomial potential of degree 4, J. Math. Phys., 52 (2011), 012702, 9 pp. doi: 10.1063/1.3544473.  Google Scholar

[13]

J. LlibreA. Mahdi and C. Valls, Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree $-2$, Phys. Lett. A, 375 (2011), 1845-1849.  doi: 10.1016/j.physleta.2011.03.042.  Google Scholar

[14]

J. Llibre and C. Valls, Darboux integrability of 2-dimensional Hamiltonian systems with homogenous potentials of degree 3, J. Math. Phys., 55 (2014), 033507, 12 pp. doi: 10.1063/1.4868701.  Google Scholar

[15]

J. Llibre and C. Valls, On the integrability of Hamiltonian systems with $d$ degrees of freedom and homogenous polynomial potential of degree $n$, Commun. Contemp. Math., 20 (2018), 1750045, 9 pp. doi: 10.1142/S0219199717500456.  Google Scholar

[16]

A. J. Maciejewski and M. Przybylska, All meromorphically integrable 2D Hamiltonian systems with homogeneous potential of degree 3, Phys. Lett. A, 327 (2004), 461-473.  doi: 10.1016/j.physleta.2004.05.042.  Google Scholar

[17]

A. J. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential, J. Math. Phys., 46 (2005), 062901, 33 pp. doi: 10.1063/1.1917311.  Google Scholar

[18]

J. J. Morales Ruiz, Differential Galois theory and non-integrability of Hamiltonian systems, vol. 179 of Progress in Mathematics, Birkhäuser Verlag, Basel, 1999. doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[19]

A. RamaniB. Dorizzi and B. Grammaticos, Painlevé conjecture revisited, Phys. Rev. Lett., 49 (1982), 1539-1541.  doi: 10.1103/PhysRevLett.49.1539.  Google Scholar

[20]

H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Phys. D, 29 (1987), 128-142.  doi: 10.1016/0167-2789(87)90050-9.  Google Scholar

[21]

H. Yoshida, A new necessary condition for the integrability of Hamiltonian systems with a two-dimensional homogeneous potential, Phys. D, 128 (1999), 53-69.  doi: 10.1016/S0167-2789(98)00313-3.  Google Scholar

[22]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, Developments in Mathematics vol. 47, Springer Natural, Singapore, 2017. doi: 10.1007/978-981-10-4226-3.  Google Scholar

[23]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I, Funktsional. Anal. i Prilozhen., 16 (1982), 30–41, 96.  Google Scholar

[24]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II, Funktsional. Anal. i Prilozhen., 17 (1983), 8-23.   Google Scholar

Table 1.  The Morales-Ramis table
Degree Eigenvalue $\lambda$ Degree Eigenvalue $\lambda$
$k$ $p+p\left(p-1\right)\frac{k}{2}$ $-3$ $\frac{25}{24}-\frac{1}{24}\left(\frac{12}{5}+6p\right)^2$
$2$ arbitrary $3$ $-\frac{1}{24}+\frac{1}{24}\left(2+6p\right)^2$
$-2$ arbitrary $3$ $-\frac{1}{24}+\frac{1}{24}\left(\frac{3}{2}+6p\right)^2$
$-5$ $\frac{49}{40}-\frac{1}{40}\left(\frac{10}{3}+10p\right)^2$ $3$ $-\frac{1}{24}+\frac{1}{24}\left(\frac{6}{5}+6p\right)^2$
$-5$ $\frac{49}{40}-\frac{1}{40}\left(4+10p\right)^2$ $3$ $-\frac{1}{24}+\frac{1}{24}\left(\frac{12}{5}+6p\right)^2$
$-4$ $\frac{9}{8}-\frac{1}{8}\left(\frac{4}{3}+4p\right)^2$ $4$ $-\frac{1}{8}+\frac{1}{8}\left(\frac{4}{3}+4p\right)^2$
$-3$ $\frac{25}{24}-\frac{1}{24}\left(2+6p\right)^2$ $5$ $-\frac{9}{40}+\frac{1}{40}\left(\frac{10}{3}+10p\right)^2$
$-3$ $\frac{25}{24}-\frac{1}{24}\left(\frac{3}{2}+6p\right)^2$ $5$ $-\frac{9}{40}+\frac{1}{40}\left(4+10p\right)^2$
$-3$ $\frac{25}{24}-\frac{1}{24}\left(\frac{6}{5}+6p\right)^2$ $k$ $\frac{1}{2}\left(\frac{k-1}{k}+p\left(p+1\right)k\right)$
Degree Eigenvalue $\lambda$ Degree Eigenvalue $\lambda$
$k$ $p+p\left(p-1\right)\frac{k}{2}$ $-3$ $\frac{25}{24}-\frac{1}{24}\left(\frac{12}{5}+6p\right)^2$
$2$ arbitrary $3$ $-\frac{1}{24}+\frac{1}{24}\left(2+6p\right)^2$
$-2$ arbitrary $3$ $-\frac{1}{24}+\frac{1}{24}\left(\frac{3}{2}+6p\right)^2$
$-5$ $\frac{49}{40}-\frac{1}{40}\left(\frac{10}{3}+10p\right)^2$ $3$ $-\frac{1}{24}+\frac{1}{24}\left(\frac{6}{5}+6p\right)^2$
$-5$ $\frac{49}{40}-\frac{1}{40}\left(4+10p\right)^2$ $3$ $-\frac{1}{24}+\frac{1}{24}\left(\frac{12}{5}+6p\right)^2$
$-4$ $\frac{9}{8}-\frac{1}{8}\left(\frac{4}{3}+4p\right)^2$ $4$ $-\frac{1}{8}+\frac{1}{8}\left(\frac{4}{3}+4p\right)^2$
$-3$ $\frac{25}{24}-\frac{1}{24}\left(2+6p\right)^2$ $5$ $-\frac{9}{40}+\frac{1}{40}\left(\frac{10}{3}+10p\right)^2$
$-3$ $\frac{25}{24}-\frac{1}{24}\left(\frac{3}{2}+6p\right)^2$ $5$ $-\frac{9}{40}+\frac{1}{40}\left(4+10p\right)^2$
$-3$ $\frac{25}{24}-\frac{1}{24}\left(\frac{6}{5}+6p\right)^2$ $k$ $\frac{1}{2}\left(\frac{k-1}{k}+p\left(p+1\right)k\right)$
Table 2.  Integers $p_0$ and $p$
Eigenvalue $3\mu$Integer $p_0$Eigenvalue $-3\mu$Integer $p$
$3\mu\in\mathcal{Z}_{-4}^1$$\frac{1}{12} \left(-4\pm3\eta\right)$$-3\mu\in\mathcal{Z}_{-4}^1$$\frac{1}{12}\left(-4\pm3\xi\right)$
$3\mu\in\mathcal{Z}_{-4}^2$$\frac{1}{4}\left(3\pm\eta\right)$$-3\mu\in\mathcal{Z}_{-4}^2$$\frac{1}{4} \left(3\pm\xi\right)$
$3\mu\in\mathcal{Z}_{-4}^3$$\frac{1}{4}\left(-2\pm\eta\right)$$-3\mu\in\mathcal{Z}_{-4}^3$$\frac{1}{4}\left(-2\pm\xi\right)$
Eigenvalue $3\mu$Integer $p_0$Eigenvalue $-3\mu$Integer $p$
$3\mu\in\mathcal{Z}_{-4}^1$$\frac{1}{12} \left(-4\pm3\eta\right)$$-3\mu\in\mathcal{Z}_{-4}^1$$\frac{1}{12}\left(-4\pm3\xi\right)$
$3\mu\in\mathcal{Z}_{-4}^2$$\frac{1}{4}\left(3\pm\eta\right)$$-3\mu\in\mathcal{Z}_{-4}^2$$\frac{1}{4} \left(3\pm\xi\right)$
$3\mu\in\mathcal{Z}_{-4}^3$$\frac{1}{4}\left(-2\pm\eta\right)$$-3\mu\in\mathcal{Z}_{-4}^3$$\frac{1}{4}\left(-2\pm\xi\right)$
Table 3.  The values of $\eta$, $\xi$ and $\mu$
Condition$(\eta, \xi)$$\mu$
$\left(3\eta, 3\xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(3, 3\right)$$0$
$\left(3\eta, \xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(3, 3\right)$$0$
$\left(\eta, 3\xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(3, 3\right)$$0$
$\left(\eta, \xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(3, 3\right)$$0$
Condition$(\eta, \xi)$$\mu$
$\left(3\eta, 3\xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(3, 3\right)$$0$
$\left(3\eta, \xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(3, 3\right)$$0$
$\left(\eta, 3\xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(3, 3\right)$$0$
$\left(\eta, \xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(3, 3\right)$$0$
Table 4.  Integers $p_0$ and $p$
Eigenvalue $\lambda_1$Integer $p_0$Eigenvalue $\lambda_2$Integer $p$
$\lambda_1\in\mathcal{Z}_{-4}^1$$\frac{1}{12} \left(-4\pm3\eta\right)$$\lambda_2\in\mathcal{Z}_{-4}^1$$\frac{1}{12}\left(-4\pm3\zeta\right)$
$\lambda_1\in\mathcal{Z}_{-4}^2$$\frac{1}{4}\left(3\pm\eta\right)$$\lambda_2\in\mathcal{Z}_{-4}^2$$\frac{1}{4} \left(3\pm\zeta\right)$
$\lambda_1\in\mathcal{Z}_{-4}^3$$\frac{1}{4}\left(-2\pm\eta\right)$$\lambda_2\in\mathcal{Z}_{-4}^3$$\frac{1}{4}\left(-2\pm\zeta\right)$
Eigenvalue $\lambda_1$Integer $p_0$Eigenvalue $\lambda_2$Integer $p$
$\lambda_1\in\mathcal{Z}_{-4}^1$$\frac{1}{12} \left(-4\pm3\eta\right)$$\lambda_2\in\mathcal{Z}_{-4}^1$$\frac{1}{12}\left(-4\pm3\zeta\right)$
$\lambda_1\in\mathcal{Z}_{-4}^2$$\frac{1}{4}\left(3\pm\eta\right)$$\lambda_2\in\mathcal{Z}_{-4}^2$$\frac{1}{4} \left(3\pm\zeta\right)$
$\lambda_1\in\mathcal{Z}_{-4}^3$$\frac{1}{4}\left(-2\pm\eta\right)$$\lambda_2\in\mathcal{Z}_{-4}^3$$\frac{1}{4}\left(-2\pm\zeta\right)$
Table 5.  The values of $\eta$, $\zeta$ and $\mu$
Condition$(\eta, \zeta)$$\mu$
$\left(3\eta, 3\zeta\right)\in \mathbb{N}\times\mathbb{N}$$\varnothing$$\varnothing$
$\left(3\eta, \zeta\right)\in \mathbb{N}\times\mathbb{N}$$\varnothing$$\varnothing$
$\left(\eta, 3\zeta\right)\in \mathbb{N}\times\mathbb{N}$$\varnothing$$\varnothing$
$\left(\eta, \xi\right)\in \mathbb{N}\times\mathbb{N}$$\varnothing$$\varnothing$
Condition$(\eta, \zeta)$$\mu$
$\left(3\eta, 3\zeta\right)\in \mathbb{N}\times\mathbb{N}$$\varnothing$$\varnothing$
$\left(3\eta, \zeta\right)\in \mathbb{N}\times\mathbb{N}$$\varnothing$$\varnothing$
$\left(\eta, 3\zeta\right)\in \mathbb{N}\times\mathbb{N}$$\varnothing$$\varnothing$
$\left(\eta, \xi\right)\in \mathbb{N}\times\mathbb{N}$$\varnothing$$\varnothing$
Table 6.  The values of $\eta$, $\zeta$ and $\mu$
Condition$(\eta, \zeta)$$\mu$
$\left(3\eta, 3\zeta\right)\in \mathbb{N}\times\mathbb{N}$$\left(5, 7\right)$ or $\left(7, 5\right)$$-\frac{2}{3}$ or $-\frac{5}{3}$
$\left(3\eta, \zeta\right)\in \mathbb{N}\times\mathbb{N}$$\left(5, 7\right)$ or $\left(7, 5\right)$$-\frac{2}{3}$ or $-\frac{5}{3}$
$\left(\eta, 3\zeta\right)\in \mathbb{N}\times\mathbb{N}$$\left(5, 7\right)$ or $\left(7, 5\right)$$-\frac{2}{3}$ or $-\frac{5}{3}$
$\left(\eta, \xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(5, 7\right)$ or $\left(7, 5\right)$$-\frac{2}{3}$ or $-\frac{5}{3}$
Condition$(\eta, \zeta)$$\mu$
$\left(3\eta, 3\zeta\right)\in \mathbb{N}\times\mathbb{N}$$\left(5, 7\right)$ or $\left(7, 5\right)$$-\frac{2}{3}$ or $-\frac{5}{3}$
$\left(3\eta, \zeta\right)\in \mathbb{N}\times\mathbb{N}$$\left(5, 7\right)$ or $\left(7, 5\right)$$-\frac{2}{3}$ or $-\frac{5}{3}$
$\left(\eta, 3\zeta\right)\in \mathbb{N}\times\mathbb{N}$$\left(5, 7\right)$ or $\left(7, 5\right)$$-\frac{2}{3}$ or $-\frac{5}{3}$
$\left(\eta, \xi\right)\in \mathbb{N}\times\mathbb{N}$$\left(5, 7\right)$ or $\left(7, 5\right)$$-\frac{2}{3}$ or $-\frac{5}{3}$
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