  We propose an efficient symbolic control synthesis algorithm for equivariant
continuous-time dynamical systems to satisfy reach-avoid specifications. The
algorithm exploits dynamical symmetries to construct lean abstractions to avoid
redundant computations during synthesis. Our proposed algorithm adds another
layer of abstraction over the common grid-based discrete abstraction before
solving the synthesis problem. It combines each set of grid cells that are at a
similar relative position from the targets and nearby obstacles, defined by the
symmetries, into a single abstract state. It uses this layer of abstraction to
guide the order by which actions are explored during synthesis over the
grid-based abstraction. We demonstrate the potential of our algorithm by
synthesizing a reach-avoid controller for a 3-dimensional ship model with
translation and rotation symmetries in the special Euclidean group SE(2).
