  We introduce a class of eccentric discs with "strong" density cusps whose
potentials are of St\"ackel form in elliptic coordinates. Our models exhibit
some striking features: sufficiently close to the location of the cusp, the
potential and surface density distribution diverge as $\Phi \propto r^{-1}$ and
$\Sigma \propto r^{-2}$, respectively. As we move outward from the centre, the
model takes a non-axisymmetric, lopsided structure. In the limit, when $r$
tends to infinity, the isocontours of $\Phi$ and $\Sigma$ become spherically
symmetric. It is shown that the configuration space is occupied by three
families of regular orbits: {\it eccentric butterfly}, {\it aligned loop} and
{\it horseshoe} orbits. These orbits are properly aligned with the surface
density distribution and can be used to construct self-consistent equilibrium
states.
