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The class of solenoidal vector fields whose lines lie in planes parallel to R² is constructed by the method of mappings. This class exhausts the set of all smooth planarhelical solutions of Gromeka's problem in some domain D ⊂ R³. In the case of domains D with cylindrical boundaries whose generators are orthogonal to R², it is shown that the choice of a specific solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonic conjugates in D² = D∩R²; i.e., Gromeka's nonlinear problem is reduced to linear boundary value problems. As an example, a specific solution of the problem for an axisymmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in D̄² in terms of wavelet systems that form bases of various spaces of functions harmonic in D². © 2011 Pleiades Publishing, Ltd.