On Efimov surfaces that are rigid 'in the small'

We consider rigid (in the class of analytic infinitesimal bendings) analytic surfaces with an isolated point of flattening and positive Gaussian curvature around this point. It is proved that such surfaces are rigid 'in the small' in the class $C^\infty$. The proof is based on the study of the asymptotic behaviour of the field of infinitesimal bending in a neighbourhood of the point of flattening and subsequent application of the techniques of the theory of generalized Cauchy–Riemann systems with a singularity in the coefficients.