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In this article, first we give a general lemma on the existence of regularhomeomorphic solutions $f$ with the hydrodynamic normalization $f(z)=z+o(1)$ as$z\to\infty$ to the degenerate Beltrami equations$\overline{\partial}f=\mu\,\partial f$ in $\mathbb C$ whose complexcoefficients $\mu$ have compact supports. On this basis, we establish criteriafor existence and representation of regular discrete open solutions for theDirichlet problem with continuous data to degenerate Beltrami equations inarbitrary simply connected bounded domains $D$ in $\mathbb C$.  Moreover, we obtain similar criteria for the existence of multi-valuedsolutions $f$ in the spirit of the theory of multi-valued analytic functions inarbitrary bounded domains $D$ in $\mathbb C$ with no boundary componentdegenerated to a single point. Note that the latter request is necessary andthat the real parts $u$ of such solutions $f$ are the so-called $A-$harmonicfunctions, i.e., single-valued continuous weak solutions of elliptice quations${\rm div}\, (A\cdot\nabla u)=0$ with matrix-valued coefficients $A$ associatedwith $\mu$. Thus, the results can be applied to potential theory in anisotropicand inhomogeneous media.

Toward the theory of Dirichlet problem for the degenerate Beltrami equations