Integrable families of Hénon-Heiles-type Hamiltonians and a new duality

Integrable Hamiltonians of type $H=\frac{1}{2}{p}_x^2+\frac{1}{2}{p}_y^2+{\mathrm{Cx}}^{\alpha +2\mathrm{}}+x^{\alpha }{y}^2$ are discussed. Using the parameters appearing in the Painlevé analysis we find that the integrable potentials for $\alpha =1\mathrm{}$ (Hénon-Heiles) and $\alpha =-\frac{2}{3}$ (Holt) are in one-to-one correspondence. Simple relations exist between the corresponding coefficients $C$, and also the dimensions of the second invariants are found to be related. Quantum integrability is also discussed.