  In the present paper we consider the Einstein-multiple-scalar field theory.
When the target space of the scalar fields is a complete, simply connected
Riemannian manifold with non-positive sectional curvature we prove that the
static and asymptotically flat solutions which posses a photon sphere are
uniquely specified by their asymptotic data, i.e. by their mass, scalar charges
and asymptotic values of the scalar fields. The unique solution with a photon
sphere and prescribed asymptotic data is explicitly constructed.
