  We consider a time-continuous branching random walk on a one-dimensional
lattice on which there is one center (lattice point) of particle generation,
called branching source. The generation of particles in the branching source is
described by a Markov branching process. Some number (finite or infinite,
depending on the problem formulation) of absorbing sources is located
symmetrically around the branching source. For such configurations of sources
we obtain necessary and sufficient conditions for the existence of an isolated
positive eigenvalue of the evolution operator. It is shown that, under some
additional assumptions, the existence of such an eigenvalue leads to an
exponential growth of the number of particles in each point of the lattice.
