  We determine hidden conformal symmetries behind the evolution equations of
black hole perturbations in a vector-tensor theory of gravity. Such hidden
symmetries are valid everywhere in the exterior region of a spherically
symmetric, asymptotically flat black hole geometry. They allow us to factorize
second order operators controlling the black hole perturbations into a product
of two commuting first order operators. As a consequence, we are able to
analytically determine the most general time-dependent solutions for the black
hole perturbation equations. We focus on solutions belonging to a highest
weight representation of a conformal symmetry, showing that they correspond to
quasi-bound states with an ingoing behaviour into the black hole horizon, and
exponential decay at spatial infinity. Their time-dependence is characterized
by purely imaginary frequencies, with imaginary parts separated by integer
numbers, as the overtones of quasi-normal modes in General Relativity.
