  We investigate semi-classical properties of Maupertuis-Jacobi correspondence
for families of 2-D Hamiltonians $(H_\lambda(x,\xi), {\cal H}_\lambda(x,\xi))$,
when ${\cal H}_\lambda(x,\xi)$ is the perturbation of a completely integrable
Hamiltonian $\widetilde{\cal H}$ veriying some isoenergetic non-degeneracy
conditions. Assuming $\hat H_\lambda$ has only discrete spectrum near $E$, and
the energy surface $\{\widetilde{\cal H}_0={\cal E}\}$ is separated by some
pairwise disjoint Lagrangian tori, we show that most of eigenvalues for $\hat
H_\lambda$ near $E$ are asymptotically degenerate as $h\to0$. This applies in
particular for the determination of trapped modes by an island, in the linear
theory of water-waves. We also consider quasi-modes localized near rational
tori. Finally, we discuss breaking of Maupertuis-Jacobi correspondence on the
equator of Katok sphere.
