On the second term in the Weyl formula for the spectrum of the Laplace operator on the two-dimensional torus and the number of integer points in spectral domains

We construct Liouville metrics on the two-dimensional torus for which the asymptotic behaviour of the second term in the Weyl formula is evaluated explicitly. We prove the instability of the second term in this formula with respect to small deformations (in the $C^1$ metric) of a Liouville metric, and establish the absence of power reduction in the Hörmander estimate on the class of closed manifolds with smooth metric in the case of integrable geodesic flow and the zero measure of the set of closed geodesics in the subspace of unit spheres of the cotangent bundle.