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

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