Darboux transforms and the Segal–Wilson construction

We consider two classes of local holomorphic solutions of the KP equation. The first class consists of all Darboux transforms of the zero solution and the second class consists of the Segal–Wilson solutions correponding to operators of rank one. We show that the second class is a (rather small) proper subset of the first class and investigate the rate of growth of this subset under the Gevrey-type generalization of the Segal–Wilson construction.