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On the equivalence of ‘non-equivalent’ algebraic realizations

Symmetries have been used with great success to determine solutions of differential equations with which they are associated. In addition to reducing the order of the equation, one can also use the Lie algebra of the symmetries to transform the equation into ‘canonical form’. The canon, in this case, is determined by the ‘standard’ realization of the Lie algebra. Following a comment in Olver P J (1995 Equivalence, Invariants and Symmetry (Cambridge: Cambridge University Press)), we conjecture that while Lie algebras may have non-equivalent realizations in the usual (point transformation) sense, all realizations of the same Lie algebra are equivalent when considered on the appropriate ordered jet space. We show how this result can have useful implications for ordinary differential equations, including linearization for equations thought to be inherently nonlinear.