Integrable quintic polynomial potential and its generalizations

We make a detailed analysis for an integrable quintic polynomial potential. By Benenti’s approach the second integral is derived, which ensures integrability. The type of separable coordinate and coordinate transformation are deduced according to a recently established systematic procedure. Bi-Hamiltonian and Lax representations are also obtained. At last we realize that it is the quintic analogue of classical Hénon–Heiles cubic system (case 1:3) and a quartic system (case 1:6:1). Another generalization is also performed.