In this paper, we present an expository account of the work done in the last few years in understanding a matrix Lax equation which arises in the study of scalar hyperbolic conservation laws with spectrally negative pure-jump Markov initial data. We begin with its extension to general N x N matrices, which is Liouville integrable on generic coadjoint orbits of a matrix Lie group. In the probabilistically interesting case in which the Lax operator is the generator of a pure-jump Markov process, the spectral curve is generically a fully reducible nodal curve. In this case, the equation is not Liouville integrable, but we can show that the flow is still conjugate to a straight line motion, and the equation is exactly solvable. En route, we establish a dictionary between an open, dense set of lower triangular generator matrices and algebro-geometric data which plays an important role in our analysis.