Contact transformations and local reducibility of ODE to the form \(y'''=0\) (Q1304726)

For third-order ODEs the action of the contact transformations \[ X=X(x,y,z), \quad Y=Y(x,y,z), \quad Z={Y_x+z \cdot Y_y\over X_x+z\cdot X_y} (z=y'), \] is not transitive. So, the equation \(y'''=0\) is transformed into the form \[ y'''=u^3 \cdot y^{\prime \prime 3}+ u^2\cdot y^{\prime \prime 2}+ u^1 \cdot y''+u^0 \] with \(u^k= u^k(x,y,y')\). The authors characterize the contact equivalence of such equation to \(y'''=0\) by four nonlinear second-order partial differential equations for the coefficient functions \(u^k\). This result is derived by computations in jet bundle terms: The above equation is identified with the section \((u^3(p), u^2(p)\), \(u^1(p), u^0(p))\) of the product bundle \(\pi :\mathbb{R}^4\times \mathbb{R}^3\to \mathbb{R}^3\), then the pseudogroup of contact transformations acts by differential prolongation on the bundles \(J^k\pi\), its orbits are investigated. As appendix are given some REDUCE procedures used for the corresponding computations.