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There is undetermined potential function $V(\phi)$ in the action of mimeticgravity which should be resolved through physical means. In generalrelativity(GR), the static spherically symmetric(SSS) solution to the Einsteinequation is a benchmark and its deformation also plays a crucial role inmimetic gravity. The equation of motion is provided with high nonlinearity, butwe can reduce primal nonlinearity to a frequent Riccati form in the SSS case ofmimetic gravity. In other words, we obtain an expression of solution to thefunctional differential equation of motion with any potential function.Remarkably, we proved rigorously that there is a zero point of first order forthe metric function $\beta(r)$ if another metric function $\alpha(r)$ possessesa pole of first order within mimetic gravity. The zero point theorem may beregarded as the generalization of Birkhoff theorem $\alpha\beta=1$ in GR. As acorollary, we show that there is a modified black hole solution for any given$V(\phi)$, which can pass the test of solar system. As another corollary, thezero point theorem provides a dynamical mechanism for the maximum size ofgalaxies. Especially, there are two analytic solutions which provide good fitsto the rotation curves of galaxies without the demand for particle dark matter.

Generalized Birkhoff theorem and its applications in mimetic gravity