  We prove that there exists a nontrivial finite energy periodic stationary
weak solution to the 3D Navier-Stokes equations (NSE). The construction relies
on a convex integration scheme utilizing new stationary building blocks
designed specifically for the NSE. The constructed family of approximate
stationary solutions is also used to prove the existence of weak solutions of
the NSE with energy profiles discontinuous on a dense set of positive Lebesgue
measure.
