I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's equations. These equations are invariant under many transformations (Lorentz transformation, rotations, spatial and temporal translations, etc. By the way is there a way, maybe from group theory, to find all the possible transformations that leaves the equation(s) invariant?) which imply as many conserved quantities thanks to Noether's theorem. In wikipedia I can see an equation that seems to give all the conserved quantities (wiki's article) but it involves the Lagrangian and I'm not sure whether the formula is valid for all systems whose Lagrangian is possible to obtain.
There is no general algorithm for doing so, and even figuring out how many conserved quantities a system has can be difficult.
A famous example is the Toda lattice. When originally proposed by Toda in 1967, this model was believed to be chaotic. It was in fact proven to be integrable (to have too many conserved quantities to be chaotic) in 1974 by Henon. See Section 3.6 of Gutzwiller's Chaos in Classical and Quantum Mechanics for more details on this story.
Traditionally, human physicists figure out conserved quantities with extensive educated guesses and strong domain knowledge. Even for those cleverest minds (e.g. Henri Poincaré), it is hard to know whether all conserved quantities have been found.
However, recent progress in machine learning provides systematic approaches to this problem. Firstly, the number of conserved quantities can be obtained by formulating the original problem to a manifold learning problem. Secondly, all conserved quantities can be discovered numerically (parameterized as neural networks). Thirdly, some of them can even be discovered symbolically.
Here are a couple of recent papers I have written on the subject: