On Schwarzschild's interior solution and perfect fluid star model

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density rho and pressure field p(r) located in a ball r leq r_0. We find a 1-parameter family of time-independent and radially symmetric solutions {(g_a, rho_a, p_a) : -2m < a < 9 kappa M/(4c^2) identifies the “physical” (i.e., such that p_a(r) geq 0 and p_a(r) is bounded in 0 leq r leq r_0) solutions {p_a : a in mathcal{U}_0} for some neighbourhood mathcal{U}_0 subset (-2m , +infty) of a = 0. For every star model {g_a : a_0 < a < a_1}, we compute the volume V(a) of the region r leq r_0 in terms of abelian integrals of the first, second, and third kind in Legendre form.