  Let $(Z,\kappa)$ be a Walsh Brownian motion with spinning measure $\kappa$.
Suppose $\mu$ is a probability measure on $\mathbb{R}^n$. We characterize all
the $\kappa$ such that $\mu$ is a stopping distribution of $(Z,\kappa)$. If we
further restrict the solution to be integrable, we show that there would be
only one choice of $\kappa$. We also generalize Vallois' embedding, and prove
that it minimizes the expectation $\mathbb{E}[\Psi(L^Z_{\tau})]$ among all the
admissible solutions $\tau$, where $\Psi$ is a strictly convex function and
$(L_t^Z)_{t \geq 0}$ is the local time of the Walsh Brownian motion at the
origin.
