Hyperbolic Monge-Ampère systems with $S_1=0$

For hyperbolic Monge-Ampère systems, a well-known solution of the equivalence problem yields two invariant tensors, $S_1$ and $S_2$, defined on the underlying $5$-manifold, where $S_2=0$ characterizes systems that are Euler-Lagrange. In this article, we consider the opposite' case, ${S}_1 = 0$, and show that the local generality of such systems is $2$ arbitrary functions of $3$ variables’. In addition, we classify all $S_1=0$ systems with cohomogeneity at most one, which turn out to be linear up to contact transformations.