  We study the existence of incompressible embeddings of surfaces into the
genus two handlebody. We show that for every compact surface with boundary,
orientable or not, there is an incompressible embedding of the surface into the
genus two handlebody. In the orientable case the embedding can be either
separating or non-separating. We also consider the case in which the genus two
handlebody is replaced by an orientable 3-manifold with a compressible boundary
component of genus greater than or equal to two.
