The problem: groups that do not contain a G2

A second order differential equation can admit up to eight Lie point symmetries: see Section 4.3 (if there really are eight symmetries, then the differential equation can be transformed into y″ = 0). The existence of a G2 of symmetries is already sufficient to make an integration procedure via line integrations possible. The more symmetries, the easier the task; so with more than two symmetries, we should expect even less trouble, should we not?

Generally speaking, we face the following problem. The group Gr, 2 < r ≤ 8, of symmetries has a certain structure expressed in the structure constants. Any integration strategy we could invent depends on this structure, and to cover all possible cases, we should make a list of all different groups that admit a realization in the x-y plane and invent a strategy for each of them. That is possible, but not really practicable: if the Gr contains a subgroup G2, then we can forget the rest, take those two symmetries, and integrate straightforwardly. Only if the Gr does not contain a subgroup G2 do we have to invent something new.