k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A - λB ∈ Γ_n if and only if φ{symbol}(A) - λφ{symbol}(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that φ{symbol} ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k-1 = 1 such that either φ{symbol}(A) = cPAP^-1 for every A ∈ M_n, or φ{symbol}(A) = cPA^TP^-1 for every A ∈ M_n.