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The present paper studies the structure of the set of stationary solutions tothe incompressible Euler equations on the rotating unit sphere that are neartwo basic zonal flows: the zonal Rossby-Haurwitz solution of degree 2 and thezonal rigid rotation $Y_1^0$ along the polar axis.  We construct a new family of non-zonal steady solutions arbitrarily close inanalytic regularity to the second degree zonal Rossby-Haurwitz stream function,for any given rotation of the sphere. This shows that any non-linear invisciddamping to a zonal flow cannot be expected for solutions near thisRossby-Haurwitz solution.  On the other hand, we prove that, under suitable conditions on the rotationof the sphere, any stationary solution close enough to the rigid rotation zonalflow $Y_1^0$ must itself be zonal, witnessing some sort of rigidity inheritedfrom the equation, the geometry of the sphere and the base flow. Nevertheless,when the conditions on the rotation of the sphere fail, the set of solutions ismuch richer and we are able to prove the existence of both explicit stationaryand travelling wave non-zonal solutions bifurcating from $Y_1^0$, in the samespirit as those emanating from the zonal Rossby-Haurwitz solution of degree 2.

On zonal steady solutions to the 2D Euler equations on the rotating unit sphere