  We introduce a notion of weak isospectrality for continuous deformations.
Consider the Laplace-Beltrami operator on a compact Riemannian manifold with
boundary with Robin boundary conditions. Given a Kronecker invariant torus
$\Lambda$ of the billiard ball map with a vector of rotation satisfying a
Diophantine condition we prove that certain integrals on $\Lambda$ involving
the function in the Robin boundary conditions remain constant under weak
isospectral deformations. To this end we construct continuous families of
quasimodes associated with $\Lambda$. We obtain also isospectral invariants of
the Laplacian with a real-valued potential on a compact manifold for continuous
deformations of the potential. As an application we prove spectral rigidity in
the case of Liouville billiard tables of dimension two.
