  We discuss a general framework of monotone skew-product semiflows under a
connected group action. In a prior work, a compact connected group $G$-action
has been considered on a strongly monotone skew-product semiflow. Here we relax
the requirement of strong monotonicity of the skew-product semiflows and the
compactness of $G$, and establish a theory concerning symmetry or monotonicity
properties of uniformly stable 1-cover minimal sets. We then apply this theory
to show rotational symmetry of certain stable entire solutions for a class of
non-autonomous reaction-diffusion equations on $\RR^n$, as well as monotonicity
of stable travelling waves of some nonlinear diffusion equations in time
recurrent structures including almost periodicity and almost automorphy.
