  There are several well-established approaches to constructing finite
difference schemes that preserve global invariants of a given partial
differential equation. However, few of these methods preserve more than one
conservation law locally. A recently-introduced strategy uses symbolic algebra
to construct finite difference schemes that preserve several local conservation
laws of a given scalar PDE in Kovalevskaya form. In this paper, we adapt the
new strategy to PDEs that are not in Kovalevskaya form and to systems of PDEs.
The Benjamin-Bona-Mahony equation and a system equivalent to the nonlinear
Schroedinger equation are used as benchmarks, showing that the strategy yields
conservative schemes which are robust and highly accurate compared to others in
the literature.
