This paper examines various known exact solutions of Einstein's field equations to assess a recent criterion [20] that must be satisfied by any static, spherically symmetric solution in hydrostatic equilibrium. The criterion is found to be fulfilled only by (i) regular solutions with vanishing surface density and pressure, and (ii) singular solutions with non-vanishing surface density. Regular solutions with non-vanishing surface density do not satisfy the criterion. The exterior Schwarzschild solution sets necessary conditions for density distributions inside a mass to yield exact solutions or equations of state compatible with general relativity. Regular solutions with finite centers and non-zero surface densities that do not satisfy the criterion cannot meet the 'actual mass' requirement of the Schwarzschild solution. The only regular solution that could be possible is the uniform (homogeneous) density distribution. The criterion [20] provides a necessary and sufficient condition for any static, spherical configuration (including core-envelope models) to be compatible with general relativity. It may be used to construct core-envelope models of stellar objects like neutron stars and to test equations of state for dense nuclear matter and relativistic stellar structures. The paper analyzes various exact solutions, including Tolman's solutions, Adler's, Adams and Cohen's, Kuchowicz's, Vaidya and Tikekar's, and Durgapal and Fuloria's solutions. It shows that solutions with vanishing surface density (Tolman's VII and Buchdahl's "gaseous" solutions) are compatible with general relativity, while solutions with non-vanishing surface density (Tolman's V and VI solutions) are not. The criterion [20] is found to be consistent with the definition of mass as 'type independence', where the mass depends only on the surface or central density, not both. The criterion provides a necessary and sufficient condition for any static spherical configuration to be compatible with general relativity. It can be used to test equations of state for dense nuclear matter and to investigate new analytic solutions and equations of state.This paper examines various known exact solutions of Einstein's field equations to assess a recent criterion [20] that must be satisfied by any static, spherically symmetric solution in hydrostatic equilibrium. The criterion is found to be fulfilled only by (i) regular solutions with vanishing surface density and pressure, and (ii) singular solutions with non-vanishing surface density. Regular solutions with non-vanishing surface density do not satisfy the criterion. The exterior Schwarzschild solution sets necessary conditions for density distributions inside a mass to yield exact solutions or equations of state compatible with general relativity. Regular solutions with finite centers and non-zero surface densities that do not satisfy the criterion cannot meet the 'actual mass' requirement of the Schwarzschild solution. The only regular solution that could be possible is the uniform (homogeneous) density distribution. The criterion [20] provides a necessary and sufficient condition for any static, spherical configuration (including core-envelope models) to be compatible with general relativity. It may be used to construct core-envelope models of stellar objects like neutron stars and to test equations of state for dense nuclear matter and relativistic stellar structures. The paper analyzes various exact solutions, including Tolman's solutions, Adler's, Adams and Cohen's, Kuchowicz's, Vaidya and Tikekar's, and Durgapal and Fuloria's solutions. It shows that solutions with vanishing surface density (Tolman's VII and Buchdahl's "gaseous" solutions) are compatible with general relativity, while solutions with non-vanishing surface density (Tolman's V and VI solutions) are not. The criterion [20] is found to be consistent with the definition of mass as 'type independence', where the mass depends only on the surface or central density, not both. The criterion provides a necessary and sufficient condition for any static spherical configuration to be compatible with general relativity. It can be used to test equations of state for dense nuclear matter and to investigate new analytic solutions and equations of state.