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The article deals with a generalized mathematical model of the dynamics oftwo point vortices in the Bose-Einstein condensate enclosed in a harmonic trap,and of the dynamics of two point vortices in an ideal fluid bounded by acircular region. In the case of a positive vortex pair, which is of interestfor physical experimental applications, a new bifurcation diagram is obtained,for which the bifurcation of four tori into one is indicated. The presence ofbifurcations of three and four tori in the integrable model of vortex dynamicswith positive intensities indicates a complex transition and the connection ofbifurcation diagrams of both limit cases. Analytical results of thispublication (the bifurcation diagram, the reduction to a system with one degreeof freedom, the stability analysis) form the basis of computer simulation ofabsolute dynamics of vortices in a fixed coordinate system in the case ofarbitrary values of the physical parameters of the model (the intensities, thevortex interaction and etc.)

On bifurcation of the four Liouville tori in one generalized integrable model of the vortex dynamics