Symmetries of (2, 3, 5)-distributions and associated Legendrian cone structures

We exploit a natural correspondence between holomorphic (2, 3, 5)-distributions and nondegenerate lines on holomorphic contact manifolds of dimension 5 to present a new perspective in the study of symmetries of (2, 3, 5)-distributions. This leads to a number of new results in this classical subject, including an unexpected relation between the multiply-transitive families of models having 7- and 6-dimensional symmetries, and a one-to-one correspondence between equivalence classes of nontransitive (2, 3, 5)-distributions with 6-dimensional symmetries and nonhomogeneous nondegenerate Legendrian curves in \({{\mathbb {P}}}^3\). An ingredient for establishing the former is an explicit classification of homogeneous nondegenerate Legendrian curves in \({{\mathbb {P}}}^3\), which we present. Moreover, our approach gives a new perspective on exceptionality of the 3 : 1 ratio for two 2-spheres rolling on each other without twisting or slipping.