We demonstrate "hidden solvability" of the nonlinear Schrödinger (NLS) equation whose nonlinearity coefficient is spatially modulated by Hermite-Gaussian functions of different order and the external potential is appropriately chosen. By means of an explicit transformation, this equation is reduced to the stationary version of the classical NLS equation, which makes it possible to use the bright and dark solitons of the latter equation to generate solitary-wave solutions in our model. Special

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... nds of explicit solutions, such as oscillating solitary waves, are analyzed in detail. The stability of these solutions is verified by means of direct integration of the underlying NLS equation. In particular, our analytical results suggest a way of controlling dynamics of solitary waves by an appropriate spatial modulation of the nonlinearity strength in Bose-Einstein condensates, through the Feshbach resonance.