Maps preserving a Ky-Fan norm of Jordan product

Given n≥2n≥2 let 2≤k≤n2≤k≤n be fixed. We study mappings ϕ   on MnMn, the algebra of n×nn×n complex matrices, that satisfy Nk(ϕ(A)ϕ(B)+ϕ(B)ϕ(A))=Nk(AB+BA)Nk(ϕ(A)ϕ(B)+ϕ(B)ϕ(A))=Nk(AB+BA) for all A, B∈MnA, B∈Mn, where Nk(C)Nk(C) denotes the Ky-Fan k-norm, the sum of k greatest singular values of the matrix C  . If n≥3n≥3, we additionally suppose either ϕ(μ1I)=μ2Iϕ(μ1I)=μ2I, for some unimodular complex μ1μ1, μ2μ2, or, that ϕ   is surjective; and when n=2n=2, the complete description is obtained without additional assumptions.