  A symmetric tensor field on a Riemannian manifold is called Killing field if
the symmetric part of its covariant derivative is equal to zero. There is a one
to one correspondence between Killing tensor fields and first integrals of the
geodesic flow which depend polynomially on the velocity. Therefore Killing
tensor fields closely relate to the problem of integrability of geodesic flows.
In particular, the following question is still open: does there exist a
Riemannian metric on the 2-torus which admits an irreducible Killing tensor
field of rank $\geq 3$? We obtain two necessary conditions on a Riemannian
metric on the 2-torus for the existence of Killing tensor fields. The first
condition is valid for Killing tensor fields of arbitrary rank and relates to
closed geodesics. The second condition is obtained for rank 3 Killing tensor
fields and relates to isolines of the Gaussian curvature.
