Cartan normal conformal connections from pairs of second-order PDEs

We explore the different geometric structures that can be constructed from the class of pairs of second-order PDEs that satisfy the condition of a vanishing generalized Wuenschmann invariant. This condition arises naturally from the requirement of a vanishing torsion tensor. In particular, we find that from this class of PDEs we can obtain all four-dimensional conformal Lorentzian metrics as well as all Cartan normal conformal O(4, 2) connections. To conclude, we briefly discuss how the conformal Einstein equations can be imposed by further restricting our class of PDEs to those satisfying additional differential conditions