Spectral surfaces for operator pairs and Hadamard matrices of F type

It is well-known that, in general, an appearance of an algebraic hypersurface of finite multiplicity in the projective joint spectrum of an operator tuple does not imply the existence of a finite-dimensional common invariant subspace.We prove that if for a pair of operators A,B the project time joint spectrum of $A, B$ and $AB$ contains the surface $\{[x,y,z,t]\in {\mathbb C}{\mathbb P}^3: x^n+y^n+(-1)^{n-1}z^n-t^n=0\}$, the under some mild conditions this implies the existence of a subspace of dimension $n$ invariant for both $A$ and $B$. Itbis shown that the appearance of this surface has a relation to complex Hadamard matrices. We give a sufficient condition for a Hadamard matrix of F type to generate such pair $A,B$. For dimensions $n=3,4,5$ where there is a complete description of comp[lex Hadamard matrices, this condition proved to be necessary as well. Finally, we prove that a pair $A,B$ such that the projective joint spectrum of $A,B,AB$ and $BA$ contains $\{ [x,y,z_1,z_2,t]\in {mathbb C}{\mathbb P}^4: x^n+y^n+(-1)^{n-1}(e^{2\pi I/n}z_1+z_2)^n-t^n=0\}$, is generated by the Fourier matrix $F_n$.