Vector distributions with very large symmetries via rational normal curves

We construct a sequence of rank 3 distributions on $n$-dimensional manifolds for any $n\geq 7$ such that the dimension of their symmetry group grows exponentially in $n$ (more precisely it is equal to $\operatorname{Fib}_{n-1}+n+2$, where $\operatorname{Fib}_n$ is the $n$-th Fibonacci number, starting with $\operatorname{Fib}_1=\operatorname{Fib}_2=1$) and such that the maximal order of weighted jet needed to determine these symmetries grows quadratically in $n$. These examples are in sharp contrast with the parabolic geometries where the dimension of a symmetry group grows polynomially with respect to the dimension of the ambient manifold and the corresponding maximal order of weighted jet space is equal to the degree of nonholonomy of the underlying distribution plus $1$. Our models are closely related to the geometry of certain curves of symplectic flags and of the rational normal curves.