  Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the
geodesic ray transform of symmetric 2-tensor fields. Let the integral of $f$
along maximal geodesics vanish on an appropriate open subset of the space of
geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it
is shown that there exists a vector field $v$ satisfying $f=dv$ on the set of
points lying on these geodesics and $v=0$ on the intersection of this set with
the boundary $\PD M$ of the manifold $M$. Using this result, a Helgason's type
of a support theorem for the geodesic ray transform is proven. The approach is
based on analytic microlocal techniques.
