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The equation of motion of a quantized vortex filament in a trappedBose-Einstein condensate [A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A {\bf62}, 063617 (2000)] has been generalized to the case of an arbitrary anharmonicanisotropic rotating trap and presented in a variational form. For condensatedensity profiles of the form $\rho=f(x^2+y^2+\mbox{Re\,}\Psi(x+iy))$ in thepresence of the plane of symmetry $y=0$, the solutions $x(z)$ describingstationary vortices of U and S types coming to the surface and solitary waveshave been found in quadratures. Analogous three-dimensional configurations ofthe vortex filament uniformly moving along the $z$ axis have also been found instrictly cylindrical geometry. The dependence of solutions on the form of thefunction $f(q)$ has been analyzed.

Hamilton's equations of motion of a vortex filament in the rotating Bose-Einstein condensate and their "soliton" solutions