  We define a new class of solutions to the WDVV associativity equations. This
class is determined by the property that one of the commuting PDEs associated
with such a WDVV solution is linearly degenerate. We reduce the problem of
classifying such solutions of the WDVV equations to the particular case of the
so-called algebraic Riccati equation and, in this way, arrive at a complete
classification of irreducible solutions.
