We consider two-dimensional relativistically invariant systems with a three-dimensional reducible configuration space and a chiral-type Lagrangian that admit higher symmetries given by polynomials in derivatives up to the fifth order. Nine such systems are known: two are Liouville-type systems, and zero-curvature representations for two others have previously been found. We here give zero-curvature representations for the remaining five systems. We show how infinite series of conservation laws can be derived from the established zero-curvature representations. We give the simplest higher symmetries; others can be constructed from the conserved densities using the Hamiltonian operator. We find scalar formulations of the spectral problems.
Demskoi, D. K., Marikhin, V. G., & Meshkov, A. G. (2006). Lax representations for triplets of two-dimensional scalar fields of the chiral-type. Theoretical and Mathematical Physics, 148(2), 1034-1048. https://doi.org/10.1007/s11232-006-0099-0