  Completely Liouville integrable Hamiltonian system with two degrees of
freedom is considered. This Hamiltonian system describes the dynamics of two
vortex filaments in a Bose-Einstein condensate enclosed in a cylindrical trap
and dynamics of two vortices in the ideal fluid contained in a domain with the
form of a circular cylinder, generatrix of which is parallel to vortices. An
explicit reduction to a system with one degree of freedom for the case of
arbitrary signs of the vortex intensities is derived from the original system
under consideration. Along with the fact that in the case of two vortices with
intensities of opposite signs, a bifurcation diagram was obtained for a
generalized two-parameter integrable family, similar to that obtained earlier
for the case of a Bose-Einstein condensate, a new bifurcation diagram was found
that had not been encountered before. Various types of vortex dynamics
corresponding to points inside chambers of bifurcation diagrams are presented.
