Four-vortex motion in the two layer Approximation: integrable case

The problem of four vortex lines with zero total circulation and zero impulse on a unlimited fluid plane, as it is known [1,3,4,16], is reduced to a problem of three point vortices and is integrated in quadratures. In the given work these results are transferred on a case of four vortices in a two-layer rotating liquid. The analysis of phase trajectories of relative motion of vortices is made, and the singularities of absolute motion on an example of a head-on, off-center collision of two two-layer vortex pairs are studied. In particular, the new class of quasistationary solutions for the given type of motions is obtained. The problems of interaction of the distributed (or finite-core) two-layer vortices are discussed.