On a Lax-type representation and separability of the anisotropic harmonic oscillator in a radial quartic potential

For the anisotropic harmonic oscillator in the radial quartic potential a Lax-type representation is given from which naturally follows the construction of independent integrals of motion. These integrals are quadratic in momenta and therefore the system of the Hamilton equations of motion is separable in generalized elliptic co-ordinates as is proved here. This system constrained to the ellipsoid is integrable too