Integrability and area-preservation property of maps

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Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. I do not assume global integrability in the sense of Liouville-Arnold - just that the phase space is foliated by invariant curves of the type $I(x)=c$, where $c$ is constant.

Must this map be necessarily area-preserving?

I do not know either how to prove this, or come up with a counter example. Similarly, for maps defined for $\mathbb{R}^{n}$...

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Nothing prevents the invariant curves from diverging away from each other. For example, $f(x,y)=(x+1,ey)$ satisfies the condition with $I(x,y)=e^{-x}y$, and magnifies areas by the factor of $e$.