This paper investigates the analytic extension of Lauricella–Saran's hypergeometric function $ F_{K} $ to symmetric domains using branched continued fractions. The authors establish new symmetric domains of analytical continuation for $ F_{K} $ under specific conditions on real and complex parameters. They use a technique to extend the convergence of branched continued fractions from a small domain to a larger domain, and apply the PC method to prove that these domains are also domains of analytical continuation. The study also discusses applicable special cases and important remarks. The key results include the convergence of branched continued fractions in symmetric domains and the analytic continuation of $ F_{K} $ in these domains. The paper also provides several theorems and corollaries that demonstrate the convergence and analytic continuation properties of the function in different domains. The authors conclude that further research is needed to develop new methods for analyzing the convergence of branched continued fractions in both partial and general cases.This paper investigates the analytic extension of Lauricella–Saran's hypergeometric function $ F_{K} $ to symmetric domains using branched continued fractions. The authors establish new symmetric domains of analytical continuation for $ F_{K} $ under specific conditions on real and complex parameters. They use a technique to extend the convergence of branched continued fractions from a small domain to a larger domain, and apply the PC method to prove that these domains are also domains of analytical continuation. The study also discusses applicable special cases and important remarks. The key results include the convergence of branched continued fractions in symmetric domains and the analytic continuation of $ F_{K} $ in these domains. The paper also provides several theorems and corollaries that demonstrate the convergence and analytic continuation properties of the function in different domains. The authors conclude that further research is needed to develop new methods for analyzing the convergence of branched continued fractions in both partial and general cases.