A class of new exact solutions is obtained for spherically symmetric and static configurations by considering a simple relation enu 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