There is undetermined potential function V ( \phi ) in the action of mimetic gravity which should be resolved through physical means .
In general relativity ( GR ) , the static spherically symmetric ( SSS ) solution to the Einstein equation is a benchmark and its deformation also plays a crucial role in mimetic gravity .
The equation of motion is provided with high nonlinearity , but we can reduce primal nonlinearity to a frequent Riccati form in the SSS case of mimetic gravity .
In other words , we obtain an expression of solution to the functional differential equation of motion with any potential function .
Remarkably , we proved rigorously that there is a zero point of first order for the metric function \beta ( r ) if another metric function \alpha ( r ) possesses a pole of first order within mimetic gravity .
The zero point theorem may be regarded as the generalization of Birkhoff theorem \alpha \beta = 1 in GR .
As a corollary , 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 , the zero point theorem provides a dynamical mechanism for the maximum size of galaxies .
Especially , there are two analytic solutions which provide good fits to the rotation curves of galaxies without the demand for particle dark matter .