  The aim of this paper is to explore the possibilities of Conley index
techniques in the study of heteroclinic connections between finite and infinite
invariant sets. For this, we remind the reader of the Poincar\'e
compactification: this transformation allows to project a $n$-dimensional
vector space $X$ on the $n$-dimensional unit hemisphere of $X\times \mathbb{R}$
and infinity on its $(n-1)$-dimensional equator called the sphere at infinity.
Under normalizability condition, vector fields on $X$ transform into vector
fields on the Poincar\'e hemisphere whose associated flows let the equator
invariant. The dynamics on the equator reflects the dynamics at infinity, but
is now finite and may be studied by Conley index techniques. Furthermore, we
observe that some non-isolated behavior may occur around the equator, and
introduce the concept of invariant sets at infinity of isolated invariant
dynamical complement. Through the construction of an extended phase space
together which an extended flow, we are able to adapt the Conley index
techniques and prove the existence of connections to such non-isolated
invariant sets.
