  We develop a Hamiltonian formulation of the Bianchi type I space-time in
conformal gravity, i.e. the theory described by a Lagrangian that is defined by
the contracted quadratic product of the Weyl tensor, in a four-dimensional
space-time. We derive the explicit forms of the super-Hamiltonian and of the
constraint expressing the conformal invariance of the theory and we write down
the system of canonical equations. To seek out exact solutions of this system
we add extra constraints on the canonical variables and we go through a global
involution algorithm which eventually leads to the closure of the constraint
algebra. The Painleve approach provides us with a proof of non-integrability,
as a consequence of the presence of movable logarithms in the general solution
of the problem. We extract all possible particular solutions that may be
written in closed analytical form. This enables us to demonstrate that the
global involution algorithm has brought forth the complete list of exact
solutions that may be written in closed analytical form. We discuss the
conformal relationship, or absence thereof, of our solutions with Einstein
spaces.
