We discuss the Riemann-Lanczos problems in two, three, and four dimensions using the theory of exterior differential systems and Janet-Riquier theory. We show that the Riemann-Lanczos problem in two dimensions is always a system in involution. For each of the two possible signatures we give the general solution in both instances and show that the occurrence of characteristic coordinates need not affect the result. In three dimensions, the Riemann-Lanczos problem is not in involution as an identity occurs. This does not prevent the existence of singular solutions and we give an example for the reduced Gödel space-time. A prolongation of this problem, whereby an integrability condition is added, leads to a prolonged system in involution. The Riemann-Lanczos problem in four dimensions is not in involution and needs to be prolongated as Bampi and Caviglia suggested. But singular solutions of it can be found and we give examples for the Gödel, Kasner, and Debever-Hubaut space-times.

We study equations of Riemann-Lanczos type on three dimensional manifolds. Obstructions to global existence for global Lanczos potentials are pointed out. We check that the imposition of the original Lanczos symmetries on the potential leads to equations which do not have a determined type, leading to problems when trying to prove global existence. We show that elliptic equations can be obtained by relaxing those symmetry requirements in at least two different ways, leading to global existence of potentials under natural conditions. A second order potential for the Ricci tensor is introduced.

We examine symmetry generators for exterior differential systems and for systems of partial differential equations and apply the Cartan theory of exterior differential systems to the Weyl-Lanczos equations and to the Lanczos wave equation in four dimensions. We look at a number of examples of symmetries for the Weyl-Lanczos equations in four dimensions and give examples of isovectors when the solution manifold is the Schwarzschild, Kasner or Gödel space-time. Solutions of the Weyl-Lanczos system are automatically solutions of the Lanczos wave equation.We give examples of symmetry generators for the Lanczos wave equation and find that they are not automatically symmetry generators for the Weyl-Lanczos equations.