  We consider the problem of a slender rod slipping along a rough surface.
Painlev\'e \cite{Painleve1895, Painleve1905a,Painleve1905b} showed that the
governing rigid body equations for this problem can exhibit multiple solutions
(the {\it indeterminate} case) or no solutions at all (the {\it inconsistent}
case), provided the coefficient of friction $\mu$ exceeds a certain critical
value $\mu_P$. Subsequently G\'enot and Brogliato \cite{GenotBrogliato1999}
proved that, from a consistent state, the rod cannot reach an inconsistent
state through slipping. Instead there is a special solution for
$\mu>\mu_C>\mu_P$, with $\mu_C$ a new critical value of the coefficient of
friction, where the rod continues to slip until it reaches a singular `$0/0$'
point $P$. Even though the rigid body equations can not describe what happens
to the rod beyond the singular point $P$, it is possible to extend the special
solution into the region of indeterminacy. This extended solution is very
reminiscent of a {\it canard} \cite{Benoit81}. To overcome the inadequacy of
the rigid body equations beyond $P$, the rigid body assumption is relaxed in
the neighbourhood of the point of contact of the rod with the rough surface.
Physically this corresponds to assuming a small compliance there. It is natural
to ask what happens to both the point $P$ and the special solution under this
regularization, in the limit of vanishing compliance. In this paper, we prove
the existence of a canard orbit in a reduced $4D$ slow-fast phase space,
connecting a $2D$ focus-type slow manifold with the stable manifold of a $2D$
saddle-type slow manifold. The proof combines several methods from local
dynamical system theory, including blowup. The analysis is not standard, since
we only gain ellipticity rather than hyperbolicity with our initial blowup.
