Killing tensor fields of third rank on a two-dimensional Riemannian torus

A rank $m$ symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero. Such a field determines the first integral of the geodesic flow which is a degree $m$ homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form $ds^2=\lambda(z)|dz|^2$ in the coordinates. The torus admits a third rank Killing tensor field if and only if the function $\lambda$ satisfies the equation $\Re\big(\frac{\partial}{\partial z}\big(\lambda(c\Delta^{-1}\lambda_{zz}+a)\big)\big)=0$ with some complex constants $a$ and $c\neq0$. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function $\lambda$. If the functions $\lambda$ and $\lambda+\lambda_0$ satisfy the equation for a real constant $\lambda_0\neq0$, then there exists a non-zero Killing vector field on the torus.