Various solutions of the kinetic equation for the equilibrium of a gravitating sphere of uniform density with a quadratic gravitational potential and a linear dependence of gravitational force on radius are examined. New analytic solutions are obtained for a uniform sphere with a hollow spherical volume and central mass inside the sphere. Solutions are also obtained for an arbitrary number of spherical layers with the same density, but with different equilibrium distribution functions.

中文翻译：

---

具有均匀密度的多层球面星团

研究了具有二次引力势和引力对半径的线性相关性的均匀密度引力球的平衡动力学方程的各种解。对于具有空心球体和球体内部中心质量的均匀球体，获得了新的解析解。对于具有相同密度但具有不同平衡分布函数的任意数量的球形层，也可以获得解。