  This paper deals with the one perturbed vortex dynamics problem which
describes the system of two point vortices in a Bose-Einstein condensate
enclosed in a cylindrical trap. This system is a completely integrable
Hamiltonian system with two degrees of freedom. Bifurcation diagram of momentum
mapping is analytically investigated. To find the bifurcation diagram we use
the method of critical subsystems developed by M.P. Kharlamov for the study of
the phase topology of integrable problems of rigid body dynamics. As an
application, we present an analysis of the stability of critical trajectories
(i.e., the nondegenerate singularities of rank 1 of the momentum mapping) by
defining a trajectory type (elliptic or hyperbolic) for each curve from the
bifurcation set.
