  We study the singular ordinary differential equation
  $$
  \frac{d U}{d t} = f (U) / z (U) + g (U),
  $$ where $U \in R^N$, the functions $f \in R^N $ and $g \in R^N $ are of
class $C^2$ and $z $ is a real valued $C^2$ function. The equation is singular
in the sense that $z (U)$ can attain the value 0. We focus on the solutions of
the singular ODE that belong to a small neighborhood of a point $\bar U$ such
that $f (\bar U) = g (\bar U) = \vec 0$, $z (\bar U) =0$. We investigate the
existence of manifolds that are locally invariant for the singular ODE and that
contain orbits with a suitable prescribed asymptotic behaviour. Under suitable
hypotheses on the set $\{U: z (U) = 0 \}$, we extend to the case of the
singular ODE the definitions of center manifold, center stable manifold and of
uniformly stable manifold. An application of our analysis concerns the study of
the viscous profiles with small total variation for a class of mixed
hyperbolic-parabolic systems in one space variable. Such a class includes the
compressible Navier Stokes equation.
