  This paper describes a new algorithm for determining all discrete contact
symmetries of any differential equation whose Lie contact symmetries are known.
The method is constructive and is easy to use. It is based upon the observation
that the adjoint action of any contact symmetry is an automorphism of the Lie
algebra of generators of Lie contact symmetries. Consequently, all contact
symmetries satisfy various compatibility conditions. These conditions enable
the discrete symmetries to be found systematically, with little effort.
