  Liouville field theory is considered with boundary conditions corresponding
to a quantization of the classical Lobachevskiy plane (i.e. euclidean version
of $AdS_2$). We solve the bootstrap equations for the out-vacuum wave function
and find an infinite set of solutions. This solutions are in one to one
correspondence with the degenerate representations of the Virasoro algebra.
Consistency of these solutions is verified by both boundary and modular
bootstrap techniques. Perturbative calculations lead to the conclusion that
only the ``basic'' solution corresponding to the identity operator provides a
``natural'' quantization of the Lobachevskiy plane.
