  In this paper we explore the phase space structures governing isomerization
dynamics on a potential energy surface with four wells and an index-2 saddle.
For this model, we analyze the influence that coupling both degrees of freedom
of the system and breaking the symmetry of the problem have on the geometrical
template of phase space structures that characterizes reaction. To achieve this
goal we apply the method of Lagrangian descriptors, a technique with the
capability of unveiling the key invariant manifolds that determine transport
processes in nonlinear dynamical systems. This approach reveals with
extraordinary detail the intricate geometry of the isomerization routes
interconnecting the different potential wells, and provides us with valuable
information to distinguish between initial conditions that undergo sequential
and concerted isomerization.
