We consider the known classes of explicitly solvable Vekua equations and construct new ones. Using the relationship between the Vekua equations and second order equations we obtain explicitly solvable stationary Schro¨dinger equations as well as explicit solutions for the equation divðruÞ ¼ 0, when is an even positive or negative power of an arbitrary harmonic function.

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A quaternionic reformulation of the Maxwell equations for chiral media is proposed. Integral representations for solutions are constructed. A complete solution of the extendability problem for the electromagnetic fields in chiral media is obtained. Maxwell's equations for inhomogeneous chiral media are studied also and some classes of solutions for slowly changing media are obtained.

We consider the static Maxwell system with an axially symmetric dielectric permittivity and construct complete systems of its solutions, which can be used for analytic and numerical solution of corresponding boundary value problems.