  The dynamics of a vortex filament in a trapped Bose-Einstein condensate is
considered when the equilibrium density of the condensate, in rotating with
angular velocity ${\bf\Omega}$ coordinate system, is Gaussian with a quadratic
form ${\bf r}\cdot\hat D{\bf r}$. It is shown that equation of motion of the
filament in the local induction approximation admits a class of exact solutions
in the form of a straight moving vortex, ${\bf R}(\beta,t)=\beta {\bf M}(t)
+{\bf N}(t)$, where $\beta$ is a longitudinal parameter, and $t$ is the time.
The vortex is in touch with an ellipsoid, as it follows from the conservation
laws ${\bf N}\cdot \hat D {\bf N}=C_1$ and ${\bf M}\cdot \hat D {\bf N}=C_0=0$.
Equation of motion for the tangent vector ${\bf M}(t)$ turns out to be closed,
and it has the integrals ${\bf M}\cdot \hat D {\bf M}=C_2$, $(|{\bf M}| -{\bf
M}\cdot\hat G{\bf \Omega})=C$, where the matrix $\hat G=2(\hat I \mbox{Tr\,}
\hat D -\hat D)^{-1}$. Intersection of the corresponding level surfaces
determines trajectories in the phase space.
