Vortices and vortical structures in systems with pairing of spatially separated electrons and holes

The properties of systems with pairing of spatially separated electrons and holes are studied below the superconducting transition temperature T_c. Another phenomenon in these systems is predicted. It is shown that in certain nonuniform magnetic fields the planar vortices in which the electron-hole pairs rotate as a whole in the plane of the structure become energy advantageous. Although the planar vortices resemble Onsager-Feynman and Abrikosov vortices, they must form a structure fundamentally different from the one the above vortices form. This structure is ordered but noninvariant under translation. In the absence of a magnetic field planar vortices appear in the system due to thermal fluctuations. The velocity field of a planar vortex decreases inversely with the distance to the vortex center. This leads to a pairing of thermally excited vortices with the opposite circulations at the temperatures below T_c.