  Dual kinetic balance (DKB) technique was previously developed to eliminate
spurious states in the finite-basis-set-based solution of the Dirac equation in
central fields. In the present paper, it is extended to the Dirac equation for
systems with axial symmetry. The efficiency of the method is demonstrated by
the calculation of the energy spectra of hydrogenlike ions in presence of
static uniform electric or magnetic fields. In addition, the DKB basis set is
implemented to solve the time-dependent Dirac equation making use of the
split-operator technique. The excitation and ionization probabilities for the
hydrogenlike argon and tin ions exposed to laser pulses are evaluated.
