  We show that the finite difference B\"acklund formula for the Schr\"odinger
Hamiltonians is a particular element of the transformation group on the set of
Riccati equations considered by two of us in a previous paper. Then, we give a
group theoretical explanation to the problem of Hamiltonians related by a first
order differential operator. A generalization of the finite difference
algorithm relating eigenfunctions of {\emph three} different Hamiltonians is
found, and some illustrative examples of the theory are analyzed, finding new
potentials for which one eigenfunction and its corresponding eigenvalue is
exactly known.
