  The present article establishes connections between the structure of the
deterministic Navier-Stokes equations and the structure of (similarity)
equations that govern self-similar solutions as expected values of certain
naturally associated stochastic cascades. A principle result is that explosion
criteria for the stochastic cascades involved in the probabilistic
representations of solutions to the respective equations coincide. While the
uniqueness problem itself remains unresolved, these connections provide
interesting problems and possible methods for investigating symmetry breaking
and the uniqueness problem for Navier-Stokes equations. In particular, new
branching Markov chains, including a dilogarithmic branching random walk on the
multiplicative group $(0,\infty)$, naturally arise as a result of this
investigation.
