Main Question or Discussion Point

Can Phase Space be break down into different regions, different regions that are not mixed up with one another, if we do so, the different regions can raise to any conservation of physical quantity?

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Simon Bridge
just consider a particle in the field with potential $V(x)=e^x+1/x$. The phase space consists of two pieces $\{(x,\dot x)\mid x>0\}$ and $\{(x,\dot x)\mid x<0\}$