  We discuss matching control laws for underactuated systems. We previously
showed that this class of matching control laws is completely charactarized by
a linear system of first order partial differential equations for one set of
variables followed by a linear system of first order PDEs for a second set of
variables. Here we derive a new first order system of partial differential
equations that encodes all compatibility conditions for the lambda-equations.
We give four examples illustrating different features of matching control laws.
The last example is a system with two unactuated degrees of freedom that admits
only basic solutions to the matching equations. There are systems with many
matching control laws where only basic solutions are potentially useful. We
introduce a rank condition indicating when this is likely to be the case.
