  In our previous work, we introduced a new class of bounded potentials of the
one-dimensional Schr\"odinger operator on the real axis, and a corresponding
family of solutions of the KdV hierarchy. These potentials, which we call
primitive, are obtained as limits of rapidly decreasing reflectionless
potentials, or multisoliton solutions of KdV. In this note, we introduce
generalized primitive potentials, which are obtained as limits of all rapidly
decreasing potentials of the Schr\"odinger operator. These potentials are
constructed by solving a contour problem, and are determined by a pair of
positive functions on a finite interval and a functional parameter on the real
axis.
