Invariance of quasi-arithmetic means with function weights

Let I⊂RI⊂R be a non-trivial interval and let λ,μ,ν:I2→(0,1). We present some results concerning the following functional equation, generalizing the Matkowski–Sutô equation,λ(x,y)φ−1(μ(x,y)φ(x)+(1−μ(x,y))φ(y))+(1−λ(x,y))ψ−1(ν(x,y)ψ(x)+(1−ν(x,y))ψ(y))=λ(x,y)x+(1−λ(x,y))y,λ(x,y)φ−1(μ(x,y)φ(x)+(1−μ(x,y))φ(y))+(1−λ(x,y))ψ−1(ν(x,y)ψ(x)+(1−ν(x,y))ψ(y))=λ(x,y)x+(1−λ(x,y))y, where φ,ψ:I→R are continuous and strictly monotonic unknown functions.