Suppose I have the following mixed linear complementarity problem: $$ \begin{align*} a+Au+Cv&=0\tag{MLCP}\\ b+Du+Bv&\geq0\\ v&\geq0\\ v^\top\left(b+Du+Bv\right)&=0 \end{align*} $$ where $a,b,A,B,C,D$ are given and $u,v$ are variables.
In the case that $A$ isn't singular this can be converted into an instance of a linear complementarity problem by taking $q=b-DA^{-1}a,M=B-DA^{-1}C$ and writing: $$ \begin{align*} z&\geq0\tag{LCP}\\ q+Mz&\geq0\\ z^\top\left(q+Mz\right)&=0 \end{align*} $$ In remark 1.7.15 of "The Linear Complementarity Problem" by Cottle, Pang, and Stone it is mentioned that the transformation of an mlcp into an lcp in the case where $A$ is singular is 'not an entirely straightforward matter' although I haven't been able to find any references for how to do this.
My question is how do I convert from an mlcp to an lcp in the case where $A$ is singular?