  We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in
1988 and then show that it can produce many of the known minimal surfaces in
$\mathbb{S}^3$ up to genus and symmetry group. We also produce several new
infinite families of minimal surfaces in $\mathbb{S}^3$ proposed by
Pitts-Rubinstein. These examples are doublings and desingularizations of
stationary integral varifolds in $\mathbb{S}^3$.
