  In close binary systems composed of a normal, donor star and an accreting
neutron star, the amount of material received by the accreting component is, so
far, a real intrigue. In the literature there are available models that link
the accretion disk surrounding the neutron star with the amount of material it
receives, but there is no model linking the amount of matter lost by the donor
star to that falling onto the neutron star.
  In this paper we explore the evolutionary response of these close binary
systems when we vary the amount of material accreted by the neutron star. We
consider a parameter \beta, which represents the fraction of material lost by
the normal star that can be accreted by the neutron star. \beta is considered
as constant throughout evolution. We have computed the evolution of a set of
models considering initial donor star masses (in solar units) between 0.5 and
3.50, initial orbital periods (in days) between 0.175 and 12, initial masses of
neutron stars (in solar units) of 0.80, 1.00, 1.20 and 1.40 and several values
of beta. We assumed solar abundances. These systems evolve to ultracompact or
to open binary systems, many of which form low mass helium white dwarfs. We
present a grid of calculations and analyze how these results are affected upon
changes in the value of \beta. We find a weak dependence of the final donor
star mass with respect to \beta. In most cases this is also true for the final
orbital period. The most sensitive quantity is the final mass of the accreting
neutron star.
  As we do not know the initial mass and rotation rate of the neutron star of
any system, we find that performing evolutionary studies is not helpful for
determining \beta.
