Integrable perturbations of the harmonic oscillator and Poisson pencils

Integrable perturbations of the two-dimensional harmonic oscillator are studied with the use of the recently developed theory of quasi-Lagrangian equations (equations of the form (q) double over dot = A(-1)(q)delk(q) where A(q) is a Killing matrix) and with the use of Poisson pencils. A quite general class of integrable perturbations depending on an arbitrary solution of a certain second-order linear PDE is found in the case of harmonic oscillator with equal frequencies. 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 Korteveg-de Vries integrable case of the Henon-Heiles system is obtained. In the case when the perturbation is of a driven type (i.e. when one of the equations is autonomous) a method of solution of these systems by separation of variables and quadratures is presented.