  We consider a third order non-autonomous ODE that arises as a model of fluid
accumulation in a two dimensional thin-film flow driven by surface tension and
gravity. With the appropriate matching conditions, the equation describes the
inner structure of solutions around a stagnation point. In this paper we prove
the existence of solutions that satisfy this problem. In order to prove the
result we first transform the equation into a four dimensional dynamical
system. In this setting the problem consists of finding heteroclinic
connections that are the intersection of a two dimensional centre-stable
manifold and a three-dimensional centre-unstable one. We then use a shooting
argument that takes advantage of the information of the flow in the far-field,
part of the analysis also requires the understanding of oscillatory solutions
with large amplitude. The far-field is represented by invariant
three-dimensional subspaces and the flow on them needs to be understood, most
of the necessary results in this regard are obtained in \cite{CV}. This
analysis focuses on the understanding of oscillatory solutions and some results
are used in the current proof, although the structure of oscillations is
somewhat more complicated.
