  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.
