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In this article, we obtain the explicit solvability of the Dirichlet problem for the Cauchy–Riemann operator, for a Beltrami operator with constant coefficient, and for the Bitsadze/Laplace operator in a lens and two complementary lunes. Using the parqueting-reflection technique, Cauchy–Pompeiu integral formulas are established. Then the expressions of both the solutions and the solvability conditions are explicitly obtained. The boundary behavior of resulting integral operators is discussed.