Abstract
In this paper we consider a system of first order nonlinear Beltrami equations with singular point in the angular unbounded region of the complex plane. This system of equations is used in the theory of surfaces of positive infinitesimal curvature with a density point and for the construction of isometric nodal coordinates on surfaces of positive curvature with a density point. In this paper, for this system of equations we obtain sufficient condition for the solution of the Dirichlet type problem in the space of continuous functions. For this purpose, we will use the general solution of the system of corresponding linear elliptic differential equations with singular point. The proof of existence of continuous solutions of the Dirichlet problem is based on the Schauder fixed point principle.