A search for integrable two-dimensional hamiltonian systems with polynomial potential

Abstract We report the results of a search for all integrable hamiltonian systems of type H = ( 1 2 )p x 2 + ( 1 2 )p y 2 + V(x,y) , where V is a polynomial in x and y of degree 5 or less and the second invariant is a polynomial in px and py of order 4 or less. Both classical and quantum integrability are discussed.