On continuous solutions of the model homogeneous Beltrami equation with a polar singularity

This paper consists of two parts. The first part is devoted to the study of the Beltrami model equation with a polar singularity in a circle centered at the origin, with a cut along the positive semiaxis. The coefficients of the equation have a first-order pole at the origin and do not even belong to the class L 2 ( G ) . For this reason, despite its specific form, this equation is not covered by the analytical apparatus of I.N. Vekua [1] and needs to be independently studied. Using the technique developed by A.B. Tungatarov [2] in combination with the methods of the theory of functions of a complex variable [3] and functional analysis [4], manifolds of continuous solutions of the Beltrami model equation with a polar singularity are obtained. The theory of these equations has numerous applications in mechanics and physics. In the second part of the article, the coefficients of the equation are chosen so that the resulting solutions are continuous in a circle without a cut [5]. These results can be used in the theory of infinitesimal bendings of surfaces of positive curvature with a flat point and in constructing a conjugate isometric coordinate system on a surface of positive curvature with a planar point [6]. equation, equation with a polar singularity.