  Here we study the global existence of `hairy' dyonic black hole and dyon
solutions to four dimensional, anti-de Sitter Einstein-Yang-Mills theories for
a general simply-connected and semisimple gauge group $G$, for so-called
topologically symmetric systems, concentrating here on the regular case. We
generalise here cases in the literature which considered purely magnetic
spherically symmetric solutions for a general gauge group and topological
dyonic solutions for $\mathfrak{su}(N)$. We are able to establish the global
existence of non-trivial solutions to all such systems, both near existing
embedded solutions and as $|\Lambda|\rightarrow\infty$. In particular, we can
identify non-trivial solutions where the gauge field functions have no zeroes,
which in the $\mathfrak{su}(N)$ case proved important to stability. We believe
that these are the most general analytically proven solutions in 4D anti-de
Sitter Einstein-Yang-Mills systems to date.
