  Extended Gross-Pitaevskii equations for the rotating F=2 condensate in a
harmonic trap are solved both numerically and variationally using trial
functions for each component of the wave function. Axially-symmetric vortex
solutions are analyzed and energies of polar and cyclic states are calculated.
The equilibrium transitions between different phases with changing of the
magnetization are studied. We show that at high magnetization the ground state
of the system is determined by interaction in "density" channel, and at low
magnetization spin interactions play a dominant role. Although there are five
hyperfine states, all the particles are always condensed in one, two or three
states. Two novel types of vortex structures are also discussed.
