  We present an algorithm which given a presentation of a group $G$ without
2-torsion, a solution to the word problem with respect to this presentation,
and an acylindricity constant ${\kappa}$, outputs a collection of tracks in an
appropriate presentation complex. We give two applications: the first is an
algorithm which decides if $G$ admits an essential free decomposition, the
second is an algorithm which; if $G$ is relatively hyperbolic; decides if it
admits an essential elementary splitting.
