  The present paper studies the structure of the set of stationary solutions to
the incompressible Euler equations on the rotating unit sphere that are near
two basic zonal flows: the zonal Rossby-Haurwitz solution of degree 2 and the
zonal rigid rotation $Y_1^0$ along the polar axis.
  We construct a new family of non-zonal steady solutions arbitrarily close in
analytic regularity to the second degree zonal Rossby-Haurwitz stream function,
for any given rotation of the sphere. This shows that any non-linear inviscid
damping to a zonal flow cannot be expected for solutions near this
Rossby-Haurwitz solution.
  On the other hand, we prove that, under suitable conditions on the rotation
of the sphere, any stationary solution close enough to the rigid rotation zonal
flow $Y_1^0$ must itself be zonal, witnessing some sort of rigidity inherited
from 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 is
much richer and we are able to prove the existence of both explicit stationary
and travelling wave non-zonal solutions bifurcating from $Y_1^0$, in the same
spirit as those emanating from the zonal Rossby-Haurwitz solution of degree 2.
