  We study the integrability of mappings obtained as reductions of the discrete
Korteweg-de Vries (KdV) equation and of two copies of the discrete potential
Korteweg-de Vries equation (pKdV). We show that the mappings corresponding to
the discrete KdV equation, which can be derived from the latter, are completely
integrable in the Liouville-Arnold sense. The mappings associated with two
copies of the pKdV equation are also shown to be integrable.
