  In this work we consider two-dimensional critical manifolds in planar
fast-slow systems near fold and so-called canard (=`duck') points. These
higher-dimension, and lower-codimension, situation is directly motivated by the
case of hysteresis operators limiting onto fast-slow systems as well as by
systems with constraints. We use geometric desingularization via blow-up to
investigate two situations for the slow flow: generic fold (or jump) points,
and canards in one-parameter families. We directly prove that the fold case is
analogous to the classical fold involving a one-dimensional critical manifold.
However, for the canard case, considerable differences and difficulties appear.
Orbits can get trapped in the two-dimensional manifold after a canard-like
passage thereby preventing small-amplitude oscillations generated by the
singular Hopf bifurcation occurring in the classical canard case, as well as
certain jump escapes.
