We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow. This situation contrasts strongly with the setting of some monotone shear flows (such as the Couette flow) and also with those of both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel. Our result has surprising consequences in the context of inviscid damping in 2D Euler and enhanced dissipation in Navier-Stokes. This is joint work with Michele Coti Zelati and Tarek Elgindi.