I am aware of results about eigenvalues of one-parameter families of Hermitian matrices sharing analytic properties with the entries of the matrix, and how this result does not extend to the two-parameter case, in particular failing at parameter values at which the spectrum is not simple (see for example "Notes on mathematics and its applications.", Rellich 1969 (Chapter 1)).
I am interested in additional restrictions to the two-parameter families which allow this theorem to extend. I consider $N \times N$ Hermitian postive-semidefinite matrices $H(a,b)$ parametrized by points on a torus $(a,b) \in \mathbb{T}^2 $. The spectrum is known to be such that the first $M$ eigenvalues are strictly greater than zero for all parameters $\lambda_1(a,b) > 0, \ldots, \lambda_M(a,b) > 0$ and the remaining $N-M$ are identically zero $\lambda_{M+1}(a,b) = \ldots = \lambda_N(a,b) = 0$. Furthermore, giving the torus coordinates $(a,b) \in [0,2\pi)\times[0,2\pi)$, the family is known to only be a function of $H(e^{ia},e^{ib})$ and in particular each entry of the matrix is a finite Laurent polynomial in these arguments (or equivalently some function of (cos(a), sin(a), cos(b), sin(b)). I also know that the principal sub-matrices of $H$ are all positive semi-definite as well.
I would like to convince myself that the non-zero eigenvalues of matrix families like this are analytic in $(a,b)$, or if that is not the case, maybe what extra conditions would $H$ need to satisfy for it to be the case.
If there are some known results about analytic eigenvalues in the case of hermitian positive semi-definite families or two-parameter periodic families, I would also appreciate any directions to relevant literature.
Many thanks in advance.