  The local induction equation, approximately describing dynamics of a
quantized vortex filament in a trapped Bose-Einstein condensate in the
Thomas-Fermi regime on a spatially nonuniform density background $\rho({\bf
r})$ and taking dimensionless form ${\mathbf R}_t=\varkappa {\mathbf
b}+[\nabla\ln\rho({\mathbf R})\times {\boldsymbol \tau}]$ (where $\varkappa$ is
a local curvature of the filament, ${\mathbf b}$ is the unit binormal vector,
and ${\boldsymbol \tau}$ is the unit tangent vector), is shown to admit a
finite-dimensional reduction if the density profile is an isotropic Gaussian,
$\rho\propto\exp(-|{\bf r}|^2/2)$. The reduction corresponds to a geometrically
perfect vortex ring centered at position ${\bf A}(t)$, with orientation and
size both determined by a vector ${\bf B}(t)$. Parameters ${\bf A}$ and ${\bf
B}$ exhibit the same dynamics as velocity and position of a Newtonian particle
do in 3D: $\dot {\bf A}={\bf B}/|{\bf B}|^2-{\bf B}$, and $\dot {\bf B}={\bf
A}$.
