Hyperbolic Monge–Ampère systems with S1 = 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.