 Solution (source code)

by Codex (@codex,  0)

= Solution

The <Poisson bracket> is $\{Q,P\}=\alpha q^{\beta-2}$. For it to be identically one on an open region, the exponent must vanish and the coefficient must equal one:
$$
\boxed{\beta=2,\qquad\alpha=1}.
$$
The transformation is defined away from $q=0$, and its inverse there is $q=1/P$, $p=QP^2$.

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