We present a set of calculations of the evolution of low-mass, solar composition stars in close binary systems together with a canonical 1.4-Msolar, neutron star (NS). We restrict the initial mass and period values to those that give rise to the formation of ultracompact systems or low-mass helium white dwarf (He WD) stars. Specifically, we computed the evolution of 40 systems for which the initial masses of the normal (donor) stars were of 1.00, 1.25, 1.50, 1.75, 2.00, 2.50, 3.00 and 3.50 Msolar, while the range of initial periods covered in this work was from 0.5 to 12 d.Calculations were performed employing the binary hydro code developed by the present authors, which handles the mass transfer rate in a fully implicit way together with state-of-the-art physical ingredients and diffusion processes. In this work we have assumed the standard scheme for the orbital evolution of the binary, considering the usual processes that produce angular momentum losses: mass loss from the system, magnetic braking and gravitational radiation. In the main part of this work we assume that the NS is able to retain only half of the matter coming from the donor star. The range of final masses has been from 0.01961 to 0.34351 Msolar and periods from 39 min to 187 d. 26 out of the 40 considered systems give rise to the formation of a He WD as a compact remnant. In performing a comparison of our results with observations, we have employed three WD-NS systems, which are among the best known ones. These are PSR J0437-4715, PSR J1012+5307 and PSR B1855+09. In order to obtain good agreement between models and observations we have had to assume that the NS is able to retain only ~10 per cent of the material released by the donor star. Otherwise, for the cases of PSR J0437-4715 and PSR B1855+09 we would be in conflict with the observed NS masses. For these two objects we have been able to fit WD and NS masses, the orbital period and also the time-scale for cooling of the WD with the evolution of one binary system. For the case of PSR J1012+5307, the problem is more delicate because the object has a mass near the threshold for the occurrence of thermonuclear flashes and the initial period is close to bifurcation point.