  We develop a Hamiltonian formulation of Bianchi type-I cosmological model in
conformal gravity, i.e. the theory described by a Lagrangian which involves the
quadratic curvature invariant constructed from the Weyl tensor, in four
dimensions. We derive the explicit forms of the super-Hamiltonian and the
constraint expressing the conformal invariance of the theory, and we write down
the system of canonical equations. To seek out exact solutions to this system
we add extra constraints on the canonical variables and we go through a global
involution algorithm that possibly leads to the closure of the constraint
algebra. This enables us to extract all possible particular solutions that may
be written in closed analytical form. On the other hand, probing the local
analytical structure we show that the system does not possess the Painleve
property (presence of movable logarithms) and that it is therefore not
integrable. We stress that there is a very fruitful interplay of local
integrability-related methods such as the Painleve test and global techniques
such as the involution algorithm. Strictly speaking, we demonstrate that the
global involution algorithm has proven to be exhaustive in the search for exact
solutions. The conformal relationship of the solutions, or absence thereof,
with Einstein spaces is highlighted.
