In this work, we investigate a class of integrable nonlinear evolution equations where extrinsic quadratic contributions are the only source of nonlinearity. Within the Inverse Scattering Transform (IST) framework and using the D-bar method based on the Riemann-Hilbert approach, we extend the analysis to systems governed purely by singular dispersion relations, and we derive integrable equations that lack intrinsic nonlinearities and are driven only by the interaction between the evolution equations and their spectral problems. By suitable choices of singular dispersion relations, we derive the "truncated" Nonlinear Schrodinger (NLS) and Korteweg-de Vries (KdV) Equations that exhibit localized coherent structures and singular asymptotic behaviors that arise independently of initial conditions. The integrability of these systems is confirmed through their associated Lax pairs. To demonstrate the physical relevance of these systems, we derive them in the context of laser-plasma interactions, where they model the formation of propagating localized structures and energy transfer dynamics.