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We derive the integrability conditions of nonautonomous nonlinearSchr$\rm\ddot o$dinger equations using the Lax Pair and SimilarityTransformation methods. We present a comparative analysis of theseintegrability conditions with those of the Painlev$\rm\acute{e}$ method. Weshow that while the Painlev$\rm\acute{e}$ integrability conditions restrict thedispersion, nonlinearity, and dissipation/gain coefficients to bespace-independent and the external potential to be only a quadratic function ofposition, the Lax Pair and the Similarity Transformation methods allow forspace-dependent coefficients and an external potential that is not restrictedto the quadratic form. The integrability conditions of thePainlev$\rm\acute{e}$ method are retrieved as a special case of our generalintegrability conditions. We also derive the integrability conditions ofnonautonomous nonlinear Schr$\rm\ddot o$dinger equations for two- andthree-spacial dimensions.

A comparative analysis of Painlev\'e, Lax Pair, and Similarity  Transformation methods in obtaining the integrability conditions of nonlinear  Schr\"odinger equations