Integrable perturbations of the harmonic oscillator and Poisson pencils

Integrable perturbations of the two-dimensional harmonic oscillator are stu died with the use of the recently developed theory of quasi-Lagrangian equa tions (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 genera l class of integrable perturbations depending on an arbitrary solution of a certain second-order linear PDE is found in the case of harmonic oscillato r with equal frequencies. For the case of nonequal frequencies all quadrati c perturbations admitting two integrals of motion which are quadratic in ve locities are found. A non-potential generalization of the Korteveg-de Vries integrable case of the Henon-Heiles system is obtained. In the case when t he perturbation is of a driven type (i.e. when one of the equations is auto nomous) a method of solution of these systems by separation of variables an d quadratures is presented.