  In this paper we consider a completely Liouville integrable Hamiltonian
system with two degrees of freedom, which describes the dynamics of two vortex
filaments in a Bose-Einstein condensate enclosed in a cylindrical trap. For
vortex pairs of positive intensity detected bifurcation of three Liouville tori
into one. Such bifurcation was found in the integrable case of
Goryachev-Chaplygin-Sretensky in the dynamics of a rigid body. For the
integrable perturbation of the physical parameter of the intensity ratio,
identified bifurcation proved to be unstable, which led to bifurcations of the
type of two tori into one and vice versa.
