Given a possibly discontinuous, bounded function f:R↦R, we consider the set of generalized flows, obtained by assigning a probability measure on the set of Carathéodory solutions to the ODE x˙=f(x). The paper provides a complete characterization of all such flows which have a Markov property in time. This is achieved in terms of (i) a positive, atomless measure supported on the set f−1(0) where f vanishes, (ii) a countable number of Poisson random variables, determining the waiting times at points in f−1(0), and (iii) a countable set of numbers θk∈[0,1], describing the probability of moving up or down, at isolated points where two distinct trajectories can originate.