  We extend a well-studied ODE model for collective behaviour by considering
anisotropic interactions among individuals. Anisotropy is modelled by limited
sensorial perception of individuals, that depends on their current direction of
motion. Consequently, the first-order model becomes implicit, and new
analytical issues, such as non-uniqueness and jump discontinuities in
velocities, are being raised. We study the well-posedness of the anisotropic
model and discuss its modes of breakdown. To extend solutions beyond breakdown
we propose a relaxation system containing a small parameter $\varepsilon$,
which can be interpreted as a small amount of inertia or response time. We show
that the limit $\varepsilon \to 0$ can be used as a jump criterion to select
the physically correct velocities. In smooth regimes, the convergence of the
relaxation system as $\varepsilon \to 0$ is guaranteed by a theorem due to
Tikhonov. We illustrate the results with numerical simulations in two
dimensions.
