  We consider Legendrian contact structures on odd-dimensional complex analytic
manifolds. We are particularly interested in integrable structures, which can
be encoded by compatible complete systems of second order PDEs on a scalar
function of many independent variables and considered up to point
transformations. Using the techniques of parabolic differential geometry, we
compute the associated regular, normal Cartan connection and give explicit
formulas for the harmonic part of the curvature. The PDE system is
trivializable by means of point transformations if and only if the harmonic
curvature vanishes identically.
  In dimension five, the harmonic curvature takes the form of a binary quartic
field, so there is a Petrov classification based on its root type. We give a
complete local classification of all five-dimensional integrable Legendrian
contact structures whose symmetry algebra is transitive on the manifold and has
at least one-dimensional isotropy algebra at any point.
