I guess according to mathematical didactic, we first think of spacetime as a set and we reason about elements of its topology and then it's furthermore equipped with a metric. Appearently it is this Riemannian metric, which people consider to be the object, which induced the minimal symmetry requirements of spacetime.
1) Regarding the relation between Riemannian geometry and the Hamiltonian formalism of classical mechanisc: Does a setting for Riemannian geometry always already imply that it's possible to cook up a symplectic structre in the cotangent bundle?
2) Are there some some more natural structures which physicists might be tempted to put on spacetime, which might then also be restricting regarding the (spacetime) symmetry structures? Is constructing quantum group symmetries (of non-commutative coordinate algebras, alla Connes?) just this?
3) I'm given a solution to a differential equation which can be thought of a resulting from a Lagrangian with a set of $n$ symmetries (e.g. $n=10$ for some spacetime models). Can this solution also be the result of a Lagrangian with fewer symmetries? Here, I'm basically asking to what extend I can reconstruct the symmetries from a solution or specific sets of the soltuon. It's kind of the inverse problem of the question "are there hidden/broken symmetries?".