Squeezed states nonlinearly generated in ring resonators are a fundamental resource in photonic quantum computing. However, as Gaussian states they are insufficient for universal quantum computation, which crucially requires non-Gaussian states. Current methods for generating non-Gaussian states are probabilistic; they involve performing photon number resolving (PNR) detection on squeezed states and postselecting specific photon-number patterns. The generation efficiency of ring resonators continues to improve with Q-engineering and the investigation of materials exhibiting stronger nonlinear responses such as indium gallium phosphide (InGaP) and aluminum gallium arsenide (AlGaAs). A more detailed analysis of states generated by highly efficient ring resonators reveals significant pump depletion and the presence of non-Gaussian states. Determining whether these deterministically generated non-Gaussian states could serve as a valuable resource and reduce reliance on PNR detection remains an open question. In this talk, I will present a theoretical approach for solving the Schrödinger equation to describe non-Gaussian states generated in integrated ring resonator systems. This approach uses asymptotic field techniques from scattering theory to characterize the light at long times and distant locations relative to the ring, where measurements are typically performed. I will provide numerical examples for InGaP ring resonators demonstrating high squeezing observed through homodyne measurements, pump depletion, and the onset of non-Gaussianity. To conclude, I will present ongoing work aimed at fully characterizing the non-Gaussian features of the state beyond the perturbative regime.