The sum of two ϕS orthogonal matrices when S−TS is normal and −1 ∉ σ(S−TS)

Let a nonsingular S∈Mn(C)S∈Mn(C) be given. For A∈Mn(C)A∈Mn(C), set ϕS(A)=S−1ATSϕS(A)=S−1ATS. We say that A   is ϕSϕSsymmetric   if ϕS(A)=AϕS(A)=A; we say that A   is ϕSϕSorthogonal   if A∈GLnA∈GLn and ϕS(A)=A−1ϕS(A)=A−1; we say that A   has a ϕSϕSpolar decomposition   if A=UPA=UP for some ϕSϕS orthogonal U   and ϕSϕS symmetric P  . Suppose that S−TSS−TS is normal and −1∉σ(S−TS)−1∉σ(S−TS). We determine conditions on A∈Mn(C)A∈Mn(C) so that A   can be written as a sum of two ϕSϕS orthogonal matrices.