  We are concerned with viscous profiles (travelling waves and steady
solutions) for mixed hyperbolic-parabolic systems in one space variable. For a
class of systems including the compressible Navier Stokes equation, these
profiles satisfy a singular ordinary differential equation in the form
\label{e:ab} dU / dt = F(U)/ z (U) . Here U takes values in $R^d$ and $F: R^d
\to R^d$ is a regular function. The real valued function $z (U)$ is as well
regular, but the equation is singular because $z (U)$ can attain the value 0.
We focus on a small enough neighbourhood of a point $\bar U$ satisfying $F(\bar
U) = \vec 0$, $z (\bar U) =0$. From the point of view of the applications to
the study of hyperbolic-parabolic systems this means restricting to systems
with small total variation. We discuss how to extend the notions of center
manifold and of uniformly stable manifold. Also, we give conditions ensuring
that if $z (U) \neq 0$ at $t=0$ then $z (U) \neq 0$ at every $t$. We provide an
example showing that if $z(U)$ becomes zero in finite time then in general the
solution $U$ of equation \eqref{e:ab} is not continuously differentiable.
