  We study the asymptotic behavior of the spectrum of a quantum system which is
a perturbation of a spherically symmetric anharmonic oscillator in dimension 2.
We prove that a large part of its eigenvalues can be obtained by
Bohr-Sommerfeld quantization rule applied to the normal form Hamiltonian and
also admit an asymptotic expansion at infinity. The proof is based on the
generalization to the present context of the normal form approach developed in
[BLM20b] (see also [PS10]) for the particular case of $T^d$.
