  This article examines the Bouton-Lie group invariants of the Navier-Stokes
equation (NSE) for incompressible fluids. Bouton's theory is applied to the
general scaling transformation admitted by the NSE and is used to derive all
self-similar solutions. In light of these, the criticality of the standard NSE
system is examined and criticality criteria are derived. The theorem of
Beale-Kato-Majda is used to rule out blow-up for a subset of Bouton's
self-similar solutions. For a subset of Leray's self-similar solutions, the
cavitation number of the fluid is found to be a scale-invariant, conserved
quantity. By extending the analysis of Bouton to higher-dimensioned manifolds,
additional conserved quantities are found, which could further elucidate the
physics of fluid turbulence.
