  In the hypothesis that the 5.4m binary RXJ0806.3+1527 consists of a low mass
helium white dwarf (donor) transferring mass towards its more massive white
dwarf companion (primary), we consider as possible donors white dwarfs which
are the result of common envelope evolution occurring when the helium core mass
of the progenitor giant was still very small (~ 0.2Msun), so that they are
surrounded by a quite massive hydrogen envelope (~1/100Msun or larger), and
live for a very long time supported by proton--proton burning. Mass transfer
from such low mass white dwarfs very probably starts during the hydrogen
burning stage, and the donor structure will remain dominated by the burning
shell until it loses all the hydrogen envelope and begins transferring helium.
We model mass transfer from these low mass white dwarfs, and show that the
radius of the donor decreases while they shed the hydrogen envelope. This
radius behavior, which is due to the fact that the white dwarf is not fully
degenerate, has two important consequences on the evolution of the binary: 1)
the orbital period decreases, with a timescale consistent with the period
decrease of the binary RXJ0806.3+1527; 2) the mass transfer rate is a factor of
about 10 smaller than from a fully degenerate white dwarf, easing the problem
connected with the small X-ray luminosity of this object. The possibility that
such evolution describes the system RXJ0806.3+1527 is also consistent with the
possible presence of hydrogen in the optical spectrum of the star, whose
confirmation would become a test of the model.
