It is known that the Bergman kernel function associated to a finitely multiply connected domain can be written as a rational combination of three holomorphic functions of one complex variable. In the case of quadrature domains, the number of generating functions reduces to two. More specifically, the Bergman kernel function associated to a quadrature domain can be written as a rational combination of only two functions, namely, the function z and the Schwarz function. In this work, we have derived analogous results for a class of special domains with symmetries. For such a domain with connectivity n, we have constructed a 2 n-tuple, which is a 2n-sheeted covering of it, and proves that the function z extends meromorphically to the 2n-tuple. We have also proved that the function z together with one branch of the Schwarz function form a primitive pair on the 2n-tuple, and consequently that the Bergman kernel function associated to this domain can be written as a rational combination of these two functions.