  In recent work on holomorphic maps that are symmetric under certain complex
reflection groups---generated by complex reflections through a set of
hyperplanes, the author announced a general conjecture related to reflection
groups. The claim is that for each reflection group G, there is a G-equivariant
holomorphic map that is critical exactly on the set of reflecting hyperplanes.
One such group is the Valentiner action V---isomorphic to the alternating group
A_6---on the complex projective plane. A previous algorithm that solved
sixth-degree equations harnessed the dynamics of a V-equivariant. However,
important global dynamical properties of this map were unproven. Revisiting the
question in light of the reflection group conjecture led to the discovery of a
degree-31 map that is critical on the 45 lines of reflection for V. The map's
critical finiteness provides a means of proving its possession of the previous
elusive global properties. Finally, a sextic-solving procedure that employs
this map's reliable dynamics is developed.
