Summary: The continuous limit for the Kac-van Moerbeke (KvM) hierarchy, for their bi-Hamiltonian formulation, recursion relation and square eigenfunction relation are studied. A new family of integrable symplectic maps (ISM) are reduced from the KvM hierarchy via constraint for a higher flow of the hierarchy in terms of square eigenfunctions. Their integrability is deduced from the discrete zero-curvature representation of the KvM hierarchy. It is shown that these ISMs provide maps which approximate many well-known integrable mechanical systems (e.g. Neumann, Garnier) embedded into the KdV hierarchy as their restricted flows.