= Solution
The <Jacobi identity> and the two brackets $\{C,H\}=\{p_\phi,H\}=0$ imply
$$
\{\{C,p_\phi\},H\}
=\{\{C,H\},p_\phi\}+\{C,\{p_\phi,H\}\}=0.
$$
Thus $\boxed{D=\{C,p_\phi\}\text{ is conserved}}$ without re-solving the motion. Explicitly $D=\partial_\phi C=p_Rp_\phi\cos\phi-(p_\phi^2/R+A)\sin\phi$. Squaring and adding the two rotated components gives
$$
\boxed{C^2+D^2=A^2+2Hp_\phi^2.}
$$
Consequently the four quantities are not functionally independent: generically there are \b[three] independent constants. One may choose, locally, $H,p_\phi$ and the orientation encoded by $C,D$; their magnitude is fixed by the displayed relation. This also matches the maximum of three independent trajectory-labeling constants for a nonstationary autonomous flow in four-dimensional phase space. On singular subsets the local functional rank can be lower: for example, at $A=0$ and $p_\phi=0$, the differentials of $C,D$ are proportional to $dp_\phi$, leaving at most two independent differentials together with $dH$; “three” is the generic count, not an assertion about every singular configuration.
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