Coupled KPZ equations and their decoupleability

We discuss characterizations of the decoupleability, partial and full, of trilinear or completely symmetric real $n\times n\times n$ tensors, which inform on the structure of certain coupled KPZ equations. Informally, when the tensor is partially decoupleable, one of the components in the coupled KPZ equation splits off from the others, while when the tensor is fully decoupleable, each of the $n$ components splits off from the others. Such a characterization is recast as a problem of membership of trilinear tensors in $O(n)$ orbits of subsets of fully decoupleable and partially decoupleable tensors. When $n=2$, we show these subsets are the same, and in this case give a single criterion in terms of the entries of a tensor for membership in the orbits of these subsets. When $n\geq 3$, the subsets are different. For $n\geq 3$, we characterize full decoupleability in terms of several abstract relations, which when $n=3$ are made explicit. When $n=3$, we also explicitly characterize partial decoupleability. The methods involve notions in applied invariant theory, relating $O(n)$ invariant subsets to stabilizer subgroup actions on smaller sets. When $n=3$ make use of the explicit basis of invariants found by Olive and Auffray. When $n=2$, we also supply two other more direct arguments.