MEROMORPHIC INTEGRABILITY OF THE HAMILTONIAN SYSTEMS WITH HOMOGENEOUS POTENTIALS OF DEGREE-4

We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian H = (p(1)(2) + p(2)(2)) /2 + 1/P(q(1) , q(2)), be -2 ing P(q(1) , q(2)) a homogeneous polynomial of degree 4 of one of the follow-ing forms +/- q(1)(4) , 4q13q2 , +/- 6q(1)(2)q(2)(2) , +/- (q(1)(2) +q(2)(2))2 , +/- q(2) (6q(12) - q(2)(2)) , +/- q(2)(2) (6q(1)(2) +q(2)(2)) , q(1)(4) + 6 mu q(1)(2)q(2)(2,) +q(1)(4) + 6 mu q(1)(2)q(2)(2) - q(2)(4) , -q(1)(4) + 6 mu q(1)(2)q(2)(2) + q(2)(4) with mu > -1/3 and mu =6 1/3, and q14 + 6 mu q(1)(2)q(2)(2) + q(2)(4) with mu not equal 6 +/- 1/3. We note that any homogeneous polynomial of degree 4 after a linear change of variables and a rescaling can be writ-ten as one of the previous polynomials. We remark that for the polynomial q(1)(4) + 6 mu q(1)(2) q(2)(2) + q(2)(4) when mu is an element of{-5/3 , -2/3} we only can prove that it has no a polynomial first integral.