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