  We investigate semi-classical properties of Maupertuis-Jacobi correspondence
in 2-D for families of Hamiltonians $(H_\lambda(x,\xi), {\cal
H}_\lambda(x,\xi))$, when ${\cal H}_\lambda(x,\xi)$ is the perturbation of
completely integrable Hamiltonian $\widetilde{\cal H}$ veriying some
isoenergetic non-degeneracy conditions. Assuming the Weyl $h$-PDO $H^w_\lambda$
has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\cal
H}={\cal E}\}$ 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.
