Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree −2

We characterize the analytic integrability of Hamiltonian systems with Hamiltonian H=12∑i=12pi2+V(q1,q2), having homogeneous potential V(q1,q2)V(q1,q2) of degree −2.

► We deal with Hamiltonian system with Hamiltonian H=(p1+p2)/2+V(q1,q2)H=(p1+p2)/2+V(q1,q2).
► V is a homogeneous polynomial of degree k  .
► Other authors did the complete integrability of these systems for k=4,3k=4,3.
► We characterize the complete integrability for k=2,1,0,−1,−2k=2,1,0,−1,−2.