Reducing degrees of freedom of a Hamiltonian by restricting first integrals

Suppose I have a Hamiltonian $H$ with a first integral $C$. Then by fixing $C=0$ is it always possible to reduce the degrees of freedom by 1?

This is typically done in the reduction of the motion in the central force field. However I cannot find any literature on extension to the general case.

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The poisson bracket needs to be non-zero right? Suppose that the poisson bracket is zero of the first integrals $C_1,C_2$. Then we can find co-ordinates $x_1=C_1$ and $x_2=C_2$ by the Caratheodory-Jacobi-Lie theorem. –  Novo Jul 4 '12 at 9:09