On integrable perturbations of harmonic oscillator

Integrable perturbations of the two-dimensional harmonic oscillator are studied with the use of the recently developed theory of quasi-Lagrangian equations. For the case of nonequal frequencies all quadratic perturbations admitting two integrals of motion which are quadratic in velocities are found. A non-potential generalization of the KdV integrable case of the Hénon—Heiles system is obtained.