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Completeness of the cubic and quartic Hénon-Heiles Hamiltonians

The quartic Hénon-Heiles Hamiltonian $H = (P_1^2+P_2^2)/2+(Ω_1 Q_1^2+Ω_2 Q_2^2)/2 +C Q_1^4+ B Q_1^2 Q_2^2 + A Q_2^4 +(1/2)(α/Q_1^2+β/Q_2^2) - γQ_1$ passes the Painlevé 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 three others are not yet integrated in the generic case $(α,β,γ)\not=(0,0,0)$. We integrate them by building a birational transformation to two fourth order first degree equations in the classification (Cosgrove, 2000) of such polynomial equations which possess the Painlevé property. This transformation involves the stationary reduction of various partial differential equations (PDEs). The result is the same as for the three cubic Hénon-Heiles Hamiltonians, namely, in all four quartic cases, a general solution which is meromorphic and hyperelliptic with genus two. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painlevé integrability (completeness property).