  We study the semi-classical trace formula at a critical energy level for a
$h$-pseudo-differential operator on $\mathbb{R}^{n}$ whose principal symbol has
a totally degenerate critical point for that energy. We compute the
contribution to the trace formula of isolated non-extremum critical points
under a condition of "real principal type". The new contribution to the trace
formula is valid for all time in a compact subset of $\mathbb{R}$ but the
result is modest since we have restrictions on the dimension.
