Valery A.Yumaguzhin,

Contact transformations and local reducibility of ODE to the form $y'''=0$.


An arbitrary contact transformation takes the ODE $y'''=0$ to the ODE of
the form
$$
  y'''=a_3(x,y,y'){y''}^3+a_2(x,y,y'){y''}^2+a_1(x,y,y')y''+a_0(x,y,y')
$$ 
Any contact transformation takes the set of ODEs of this form
to itself. It means that the pseudogroup of all contact
transformation acts on the bundle of ODEs of this form. This
action is lifted in natural way to the jet bundles of this bundle.
We investigate the orbits of this lifted actions. In this way, we
obtain that equations defining the orbit of minimal dimension in
2-jet bundle are relations for the right-hand side of an arbitrary
3-order ODE necessary and sufficient for the existence of a
contact transformation reducing this equation to the form y'''=0$.