  We study vortical states in a Bose-Einstein condensate (BEC) filling a
cigar-shaped trap. An effective one-dimensional (1D) nonpolynomial Schroedinger
equation (NPSE) is derived in this setting, for the models with both repulsive
and attractive inter-atomic interactions. Analytical formulas for the density
profiles are obtained from the NPSE in the case of self-repulsion within the
Thomas-Fermi approximation, and in the case of the self-attraction as exact
solutions (bright solitons). A crucially important ingredient of the analysis
is the comparison of these predictions with direct numerical solutions for the
vortex states in the underlying 3D Gross-Pitaevskii equation (GPE). The
comparison demonstrates that the NPSE provides for a very accurate
approximation, in all the cases, including the prediction of the stability of
the bright solitons and collapse threshold for them. In addition to the
straight cigar-shaped trap, we also consider a torus-shaped configuration. In
that case, we find a threshold for the transition from the axially uniform
state, with the transverse intrinsic vorticity, to a symmetry-breaking pattern,
due to the instability in the self-attractive BEC filling the circular trap.
