Spectral hypersurfaces for operator pairs and Hadamard matrices of F type

We prove that if for a pair of n× n matrices A and B, with A being normal, the projective joint spectrum of A, B, AB and the identity is given by σ(A,B,AB,I)={[x,y,z,t]∈CP3:xn+yn+(-1)n-1zn-tn=0},then this pair is unitary equivalent to a one associated with a complex Hadamard matrix of order n. If n= 3 , 4 , or 5, where there is a complete description of Hadamard matrices, we list those that generate a pair with the above mentioned spectrum. If σ(A,B,AB,BA,I)={[x,y,z1,z2,t]∈CP4:xn+yn+(-1)n-1(e2πi/nz1+z2)n-tn=0},this Hadamard matrix is exactly the Fourier matrix F_n. If for an operator pair A, B acting on a Hilbert space, such hypersurfaces appear as components of the projective joint spectrum of the corresponding tuples, then under some mild conditions the pair has a common invariant subspace of dimension n, and the restriction of A, B to this subspace is generated by a Hadamard matrix of F type.