The Cauchy problem for second-order elliptic systems on the plane

Regular solutions to second-order elliptic systems on the plane are representable in terms of $A$-analytic functions satisfying an operator equation of the Beltrami type. We prove Carleman-type formulas for reconstruction of solutions from data on a part of the boundary of the domain. We use these formulas for solving the Cauchy problems for the system of Lame equations, the Navier–Stokes system, and the system of equations of elasticity with resilience.