The authors study the non-degenerate \(n\)-connected canonical domains with \(n > 1\) related to the conjecture of \textit{S.~Bell} [Duke Math. J. 98, 187--207 (1999; Zbl 0948.30015)]. These domains are connected to the algebraic property of the Bergman kernel and the Szegö kernel. S. Bell posed the following problem while he was seeking the domain with algebraic Bergman kernel: Can every non-degenerate \(n\)-connected planar domain with \(n > 1\) be mapped biholomorphically onto a domain of the form \[ \left\{z\in\mathbb C:\left| z+\sum_{k=1}^{n-1}\frac{a_ k}{z-b_ k}\right|