Completeness of the Cubic and Quartic Henon–Heiles Hamiltonians

The quartic Henon–Heiles Hamiltonian passes the Painleve test for only four sets of values of the constants. Only one of these, identical to the traveling-wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the other three have not yet been integrated in the general case $(\alpha,\beta,\gamma)\neq(0,0,0)$. We integrate them by building a birational transformation to two fourth-order first-degree equations in the Cosgrove classiffication of polynomial equations that have the Painleve property. This transformation involves the stationary reduction of various partial differential equations. The result is the same as for the three cubic Henon–Heiles Hamiltonians, namely, a general solution that is meromorphic and hyperelliptic with genus two in all four quartic cases. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painleve integrability (the completeness property).