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.