A class of new exact solutions is obtained for spherically symmetric and static configurations by considering a simple relation e^nu varies as (1+x)ⁿ. For each integral value of n the field equations can be solved exactly and one gets a new exact solution. For physical relevance of the solutions, the pressure and the density should be finite and positive and the density, P/ rho and dP/d rho should decrease as one goes outwards from the centre to the surface of the structure. Most of the exact solutions known at present are irregular in this respect. The new exact solutions for n=3, 4 and 5 are regular in this respect for a certain range of values of u(=mass/radius). The cases corresponding to n=1 and 2 are already available in the literature, being obtained by other methods. For regular solutions with dP/d rho <or=1, the maximum values of the surface and central redshifts are 0.635 and 1.614 respectively. If one assumes the surface density to be 2*10¹⁴ g cm^-3, a neutron star model corresponding to a mass up to 4.2 M_(.) can be obtained. This is an upper limit for a neutron star model based upon exact solutions with completely regular behaviour and dP/d rho <or=1. In the limiting case when dP/d rho is infinite, the surface and the central redshifts are 1.14 and 7.36 respectively. The variation of density is slow, and for a completely regular solution the maximum value for the ratio of the central to surface densities, that is rho ₀/ rho _s, is 3.0.