  Whether the 3D incompressible Navier-Stokes equations will have a global
smooth solution for all smooth, finite energy initial data is a Millennium
Prize problem. One of the main difficulties of this problem is that the
Navier-Stokes equations are actually a system of semilinear heat equations
rather than a single equation. In this paper, we discover a remarkable hidden
symmetry of the 3D incompressible Navier-Stokes equations. Under this symmetric
reduction, the system reduces to a single scalar semilinear heat equation. The
symmetry also holds for the 3D incompressible Euler equations.
