Numerical experiments have been performed to investigate the thermal behavior of a cooled down white dwarf of initial mass $M_{rm WD} = 0.516 M_{sun}$ which accretes hydrogen-rich matter with Z = 0.02 at the rate $dot{M}=10^{-8}$ msun yrm1, typical for a recurrent hydrogen shell flash regime. The evolution of the main physical quantities of a model during a pulse cycle is examined in detail. From selected models in the mass range $M_{rm WD} = 0.52div 0.68$ msunend, we derive the borders in the $M_{rm WD}$ - $dot{M}$ plane of the steady state accretion regime when hydrogen is burned at a constant rate as rapidly as it is accreted. The physical properties during a hydrogen shell flash in white dwarfs accreting hydrogen-rich matter with metallicities Z = 0.001 and Z = 0.0001 are also studied. For a fixed accretion rate, a decrease in the metallicity of the accreted matter leads to an increase in the thickness of the hydrogen-rich layer at outburst and a decrease in the hydrogen-burning shell efficiency. In the $M_{rm WD}$-$dot{M}$ plane, the borders of the steady state accretion band are critically dependent on the metallicity of the accreted matter: on decreasing the metallicity, the band is shifted to lower accretion rates and its width in $dot{M}$ is reduced.
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