The ϕS polar decomposition when the cosquare of S is normal

Let S∈Mn(C) be nonsingular such that S−TS is normal (that is, the cosquare of S is normal). Set ϕS(A)=S−1ATS for A∈Mn(C). We determine conditions on A so that A has a ϕS polar decomposition. We also find the possible Jordan Canonical Forms of a ϕS orthogonal matrix and of a ϕS skew symmetric matrix in the cases (a) 1∉σ(S−TS) and (b) −1∉σ(S−TS).