The Tolman-Oppenheimer-Volkov (TOV) equation is solved with a new ansatz: the external boundary condition with mass M0 and radius R1 is dual to the internal boundary condition with density ρbc and inner radius ri, and the two boundary conditions yield the same result. The inner boundary condition is imposed with a density ρbc and an inner radius ri, which is zero for the compact neutron stars, but non-zero for the shell-stars: stellar shell-star and galactic (supermassive) shell-star. Parametric solutions are calculated for neutron stars, stellar shell-stars, and galactic shell-stars. From the results, an M-R-relation and mass limits for these star models can be extracted. A new method is found for solving the Einstein equations for Kerr space-time with matter (extended Kerr space-time), i.e. rotating matter distribution in its own gravitational field. Then numerical solutions are calculated for several astrophysical models: white dwarf, neutron star, stellar shell-star, and galactic shell-star. The results are that shell-star star models closely resemble the behaviour of abstract black holes, including the Bekenstein-Hawking entropy, but have finite redshifts and escape velocity v < c and no singularity.
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