Integrability of generalized Jacobi problem

We consider a point moving in an ellipsoid $a_1 x_1^2 + a_2 x_2^2 + a_3 x_3^2 = 1$ under the influence of a force with quadratic potential $V=\frac{1}{2} (b_1 x_1^2 + b_2 x_2^2 + b_3 x_3^2)$. We prove that the equations of motion of the point are meromorphically integrable if and only if the condition $b_1 (a_2 - a_3) + b_2 (a_3 - a_1) + b_3 (a_1 - a_2) = 0$ is fulfilled.