  We study via Monte Carlo simulation a generalisation of the so-called vertex
interacting self-avoiding walk (VISAW) model on the square lattice. The
configurations are actually not self-avoiding walks but rather restricted
self-avoiding trails (bond avoiding paths) which may visit a site of the
lattice twice provided the path does not cross itself: to distinguish this
subset of trails we shall call these configurations \emph{grooves}. Three
distinct interactions are added to the configurations: firstly the VISAW
interaction, which is associated with doubly visited sites, secondly a nearest
neighbour interaction in the same fashion as the canonical interacting
self-avoiding walk (ISAW) and thirdly, a stiffness energy to enhance or
decrease the probability of bends in the configuration.
  In addition to the normal high temperature phase we find three low
temperature phases: (i) the usual amorphous liquid drop-like "globular" phase,
(ii) an anisotropic "$\beta$-sheet" phase with dominant configurations
consisting of aligned long straight segments, which has been found in
semi-flexible nearest neighbour ISAW models, and (iii) a maximally dense phase,
where the all sites of the path are associated with doubly visited sites
(except those of the boundary of the configuration), previously observed in
interacting self-avoiding trails.
  We construct a phase diagram using the fluctuations of the energy parameters
and three order parameters. The $\beta$-sheet and maximally dense phases do not
seem to meet in the phase space and are always separated by either the extended
or globular phases. We focus attention on the transition between the extended
and maximally dense phases, as that is the transition in the original VISAW
model. We find that for the path lengths considered there is a range of
parameters where the transition is first order and it is otherwise continuous.
