  Time independent Hamiltonians of the physical type H =
(P_1^2+P_2^2)/2+V(Q_1,Q_2) pass the Painleve' test for only seven potentials
$V$, known as the He'non-Heiles Hamiltonians, each depending on a finite number
of free constants. Proving the Painleve' property was not yet achieved for
generic values of the free constants. We integrate each missing case by
building a birational transformation to some fourth order first degree ordinary
differential equation in the classification (Cosgrove, 2000) of such polynomial
equations which possess the Painleve' property. The properties common to each
Hamiltonian are: (i) the general solution is meromorphic and expressed with
hyperelliptic functions of genus two, (ii) the Hamiltonian is complete (the
addition of any time independent term would ruin the Painleve' property).
