Suppose I have a gradient field $$\dot{x} = \nabla F(x)$$ and suppose there is a constant of motion $G(x) = c$. How do results on $ \dot{x} = \nabla (F(x) + G(x))$ connect back to the original system, $\dot{x} = \nabla F(x)$?
In the problem I am looking at $\dot{x} = \nabla F(x)$ has a fixed point for which it is difficult to determine stability (because of parameters). But $ \dot{x} = \nabla (F(x) + G(x))$ has a fixed point whose stability is easy to determine. However I am not sure how to return to the original system.