  The key problem of the theory of exterior differential systems (EDS) is to
decide whether or not a system is in involution. The special case of EDSs
generated by one-forms (Pfaffian systems) can be adequately illustrated by a
2-dimensional example. In 4 dimensions two such problems arise in a natural
way, namely, the Riemann-Lanczos and the Weyl-Lanczos problems. It is known
from the work of Bampi and Caviglia that the Weyl-Lanczos problem is always in
involution in both 4 and 5 dimensions but that the Riemann-Lanczos problem
fails to be in involution even for 4 dimensions. However, singular solutions of
it can be found. We give examples of singular solutions for the Goedel, Kasner
and Debever-Hubaut spacetimes. It is even possible that the singular solution
can inherit the spacetime symmetries as in the Debever-Hubaut case. We comment
on the Riemann-Lanczos problem in 5 dimensions which is neither in involution
nor does it admit a 5-dimensional involution of Vessiot vector fields in the
generic case.
