We present the expressions of the generalized Ginzburg-Landau (GL) theory for the free energy and the supercurrent in terms of the d vector, the magnetic field H, and operators containing the spatial gradient and the rotation Ω. These expressions are then specialized to the Anderson-Brinkman-Morel (ABM) state. We consider eight single-vortex lines of cylindrical symmetry and radius R=[2 mΩ/ℏ]-1/2: the Mermin-Ho vortex, a second analytic vortex, and six singular vortices, i.e., the orbital and radial disgyrations, the orbital and radial phase vortices, and two axial phase vortices. These eight vortex states are determined by solving the Euler-Lagrange equations whose solutions minimize the GL free energy functional. For increasing field, the core radius of the {Mathematical expression} texture of the Mermin-Ho vortex tends to a limiting value, while the core radius of the {Mathematical expression} texture goes to zero. The gap of the singular vortices behaves like rα for r → 0, where α ranges between {Mathematical expression} and {Mathematical expression}. The energy of the radial disgyration becomes lower than that of the Mermin-Ho vortex for fields H≥6.5 H*=6.5×25 G (at T=0.99 Tc and for R=10 L*=60 μm, or ω=2.9 rad/sec). For R → 2ξT(ξT is the GL coherence length) or ω →ωc2 (upper critical rotation speed), the energies of the singular vortices become lower than the energies of the analytic vortices. This is in agreement with the exact result of Schopohl for a vortex lattice at Ωc2. Finally, we calculate the correction of order (1 -T/Tc) to the GL gap for the axial phase vortex. © 1982 Plenum Publishing Corporation.