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Tanaka's equation on the circle and stochastic flows

We define a Tanaka's equation on an oriented graph with two edges and two vertices. This graph will be embedded in the unit circle. Extending this equation to flows of kernels, we show that the laws of the flows of kernels $K$ solution of Tanaka's equation can be classified by pairs of probability measures $(m^+,m^-)$ on $[0,1]$, with mean 1/2. What happens at the first vertex is governed by $m^+$, and at the second by $m^-$. For each vertex $P$, we construct a sequence of stopping times along which the image of the whole circle by $K$ is reduced to $P$. We also prove that the supports of these flows contains a finite number of points, and that except for some particular cases this number of points can be arbitrarily large.