 Solution

The Jacobi identity and the two brackets $\{C,H\}=\{p_{\varphi },H\}=0$ implyThus $\boxed{D=\{C,p_{\varphi }\}\text{ is conserved}}$ without re-solving the motion. Explicitly $D={\partial }_{\varphi }C=p_R{p}_{\varphi }\cos \varphi -(p_{\varphi }^2/R+A)\sin \varphi$. Squaring and adding the two rotated components givesConsequently the four quantities are not functionally independent: generically there are three independent constants. One may choose, locally, $H,p_{\varphi }$ 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 $p_{\varphi }=0$, the differentials of $C,D$ are proportional to $d{p}_{\varphi }$, leaving at most two independent differentials together with $dH$; “three” is the generic count, not an assertion about every singular configuration.