Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4

<p style='text-indent:20px;'>We characterize the meromorphic Liouville integrability of Hamiltonian systems with <inline-formula><tex-math id="M2">\begin{document}$ H = \left(p_1^2+p_2^2\right)/2+1/P(q_1, q_2) $\end{document}</tex-math></inline-formula>, being id="M3">\begin{document}$ P(q_1, $\end{document}</tex-math></inline-formula> a homogeneous polynomial degree id="M4">\begin{document}$ 4 one following forms id="M5">\begin{document}$ \pm q_1^4 id="M6">\begin{document}$ 4q_1^3q_2 id="M7">\begin{document}$ 6q_1^2q_2^2 id="M8">\begin{document}$ \left(q_1^2+q_2^2\right)^2 id="M9">\begin{document}$ q_2^2\left(6q_1^2-q_2^2\right) id="M10">\begin{document}$ q_2^2\left(6q_1^2+q_2^2\right) id="M11">\begin{document}$ q_1^4+6\mu q_1^2q_2^2-q_2^4 id="M12">\begin{document}$ -q_1^4+6\mu q_1^2q_2^2+q_2^4 id="M13">\begin{document}$ \mu>-1/3 and id="M14">\begin{document}$ \mu\neq 1/3 id="M15">\begin{document}$ id="M16">\begin{document}$ \mu \neq $\end{document}</tex-math></inline-formula>. We note that any id="M17">\begin{document}$ after linear change variables rescaling can be written as previous polynomials. remark for id="M18">\begin{document}$ when id="M19">\begin{document}$ \mu\in\left\{-5/3, -2/3\right\} we only prove it has no first integral.</p>