Canonical representative of a ${\mathcal{C}}^{\mathrm{\infty }}$-symmetry

For a second-order ODE, any $\lambda$-symmetry $v=\xi {\partial }_x+\eta {\partial }_u$ can be locally transformed into a simplified vertical form known as its canonical representative:

Transformation Formula

The canonical representative ${\partial }_u$ is a $\tilde{\lambda }$-symmetry, where the new function $\tilde{\lambda }$ is derived from the original $\lambda$ and the characteristic $Q=\eta -u_1\xi$:

(Here $A$ is the total derivative operator or the vector field associated to the ODE).

Utility

Using the canonical form reduces the problem of finding a symmetry from determining two functions ($\xi ,\eta$) to finding a single function $\tilde{\lambda }$. This drastically simplifies the determining equations and allows for the use of specific ansatzes (e.g., polynomial forms) to classify and solve ODEs.