I adapt the Generalised Method of Moments to deal with nonlinear models in which a finite number of isolated parameter values satisfy the moment conditions. To do so, I initially study the closely related limiting class of first-order underidentified models, whose expected Jacobian is rank deficient but not necessarily 0. In both cases, the proposed procedures yield efficiency gains and underidentification tests within a standard asymptotic framework. I study models with and without separation of data and parameters. Finally, I illustrate the proposed inference procedures with a dynamic panel data model and a non-linear regression model for discrete data.