Laplace maps and constraints for a class of third order partial differential operators

We explore the existence of a class of generalised Laplace maps for third order partial differential operators of the form $$\partial_1\partial_2\partial_3+a_1\partial_2\partial_3+a_2\partial_1\partial_3+a_3\partial_1\partial_2+a_{12}\partial_3+a_{23}\partial_1+a_{13}\partial_2+a_{123}$$ and related first order $3\time 3$ systems and show that they require the satisfaction of constraints on the invariants for such operators.