A Monster Tower approach to Goursat multi-flags

We consider the problem of classifying the orbits within a tower of fibrations with P2P2-fibers that generalize the Monster Tower due to Montgomery and Zhitomirskii. The action on the tower is given by prolongations of diffeomorphism germs of 3-space. As a corollary we give the first steps towards the problem of classifying Goursat 2-flags of small length. In short, we classify the orbits within the first four levels of the Monster Tower and show that there is a total of 34 orbits in the fourth level of the tower.