My problem comes from physics, and I am mainly interested in finding what mathematical term or framework exist that deals with it.
Motivation
I have a Hamiltonian classical $H({q_k,p_k})$ which depends on a set of $6N$ coordinates and momenta $\{q_k,p_k\}$, so it does not depend implicitly on the time.
The form of the hamiltonian is a finite sum of identical functions which depend only on a subset of coordinates $\{q_k\}_l$ or on a subset of momenta $\{p_k\}_l$: $A_l(\{q_k\}_l)$ and $B_l(\{p_k\}_l)$ which is concretely:
$$H({q_k,p_k}) = \sum_l T(B_l) + U(A_l)$$
Where $T$ and $U$ are known and I will refer to the respectively as ''Kinetic'' and ''Potential'' terms, but for which I prefer to leave the form unspecified for more generality.
This shape makes it very tempting to transform to the new coordinates and momenta $A_l$ and $B_l$, but they exceed the number of original independent canonical variables $6N$, which is reflected in the fact that they do not commute.
Particularly their commutators yield:
$$\{A_i,A_j\} = \sum_l \left[ \frac{\partial A_i}{\partial q_k} \frac{\partial A_j}{\partial p_l} - \frac{\partial A_i}{\partial p_l} \frac{\partial A_j}{\partial q_l} \right]=0$$
$$\{B_i,B_j\} = \sum_l \left[ \frac{\partial B_i}{\partial q_k} \frac{\partial B_j}{\partial p_l} - \frac{\partial B_i}{\partial p_l} \frac{\partial B_j}{\partial q_l} \right]=0$$
$$\{A_i,B_j\} = \sum_l \left[ \frac{\partial A_i}{\partial q_k} \frac{\partial A_j}{\partial p_l} - \frac{\partial A_i}{\partial p_l} \frac{\partial A_j}{\partial q_l} \right]=C_{ij}$$
with $C_{ij}$ some constant independent of any of $\{q_k,p_k\}$.
If $C_{ij} = \delta_{ij}$ the transformation would be canonical and the system could be solved in the new variables, since the hamiltonian can produce the same motion equations for the new variables, or at least some new hamiltonian which can be found, and for which methods are available.
My problem here is that this is not the case and $C_{ij}$ are not as required.
The question is then
I understand that since $\{A_l,B_l\}$ are more than $6N$ the new hamilton equations are not all independent. But I am wondering if this non canonical transformation could still be employed.
This would be the equivalent of expressing my $6N$-dimensional function in a higher amount of variables, which not all of them are independent. Then I would have an overdetermined system of differential equations, which is simpler to solve (if I could integrate each equation independently of the others) and then consider their interdependence when using the initial conditions to find all the constants of integration appearing, which of course will be larger, but the relations between them are known from the relations between the $\{A_l,B_l\}$.
So I guess I am asking for a theory on overdetermined system of differential equations and/or a theory of non-commutation transformations in classical mechanics.