I've been trying to understand how Lie Groups can help solve differential equations (this 12-page pdf is the most straightforward explanation I've seen).
My understanding is that when a first-order differential equation $\frac{dy}{dx}=h(x,y)$ has a continuous "translation symmetry" $(x,y)\mapsto(x,y+\lambda)$, it becomes very easy to solve the equation using basic integration of one variable.
The problem is that most differential equations in standard $(x,y)$ coordinates don't have this nice translation symmetry, and so we need to change to a set of "canonical coordinates" $(X,Y)$ which do have the desired translation symmetry $(X,Y)\mapsto(X,Y+\lambda)$.
The main condition that these new canonical coordinates need to satisfy, is that the $Y$ coordinate should be aligned with the $\lambda$ variable ($\frac{dY}{d\lambda}=1,\frac{dX}{d\lambda}=0$), so that we get the nice translation symmetry we want:
$$\frac{dY}{d\lambda}=\frac{dY}{dx}\frac{dx}{d\lambda}+\frac{dY}{dy}\frac{dy}{d\lambda}=Y_x\xi+Y_y\eta=1$$
$$\frac{dX}{d\lambda}=\frac{dX}{dx}\frac{dx}{d\lambda}+\frac{dX}{dy}\frac{dy}{d\lambda}=X_x\xi+X_y\eta=0$$
We also require that the symmetry condition be satisfied:
$$\frac{dY}{dX}=\frac{D_xY}{D_xX}=\frac{Y_x+Y_y y'}{X_x+X_y y'}=h(X,Y)$$
Here's where I'm stuck. Basically all the terms in the above equations are unknowns, and it seems incredibly difficult to solve for the new coordinates $(X,Y)$. I've also seen the "linearized symmetry condition" below, but it still results in a difficult-to-solve equation:
$$\eta_x-\xi_y h^2 +(\eta_y -\xi_x)h - (\xi h_x + \eta h_y) = 0$$
Is there a nice, algorithmic approach for deducing a set of canonical coordinates? Or does it ultimately come down to "guess and check" with the above equation?