We consider a porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficient with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. %This equation is motivated by some singular %behaviour arising in complex self-organized critical systems. The interest in such a singular porous media equations is due to the fact that they can model systems exhibiting the phenomenon of self-organized criticality. One of the main analytic ingredients of the proof is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients.