  Lagrangian coherent structures (LCSs) are material surfaces that shape
finite-time tracer patterns in flows with arbitrary time dependence. Depending
on their deformation properties, elliptic and hyperbolic LCSs have been
identified from different variational principles, solving different equations.
Here we observe that, in three dimensions, initial positions of all variational
LCSs are invariant manifolds of the same autonomous dynamical system, generated
by the intermediate eigenvector field, $\xi_{2}(x_{0})$, of the Cauchy-Green
strain tensor. This $\xi_{2}$-system allows for the detection of LCSs in any
unsteady flow by classic methods, such as Poincar\'e maps, developed for
autonomous dynamical systems. As examples, we consider both steady and
time-aperiodic flows, and use their dual $\xi_{2}$-system to uncover both
hyperbolic and elliptic LCSs from a single computation.
