<p>We extend the exact quadratic-profile solution of Parlange et al. (Transp Porous Media 39(3):339–345, 2000) to the generalized Boussinesq equation with a general exponent. This broader formulation is commonly used to describe saturated flow in unconfined aquifers, where hydraulic conductivity depends strongly on water-table height and nonlinear diffusion controls how wetting and drainage fronts move through the subsurface. Situations such as bank storage during river stage fluctuations, infiltration from surface sources, and drainage in shallow groundwater systems all fall within this setting and benefit from analytical solutions that capture these nonlinear effects. This exponent controls the degree of nonlinearity in the diffusion term and allows the model to represent a wide range of physical situations, from nearly linear flow to a strongly nonlinear transport, such as flows through concretes, forest soils, and filtration of polytropic gases. By extending the quadratic-profile approach to arbitrary nonlinearity, we obtain a family of exact solutions that describes both infiltration and drainage regimes within a single framework. The solution family also connects directly to two classical results: a traveling-front profile that appears under strong boundary fluxes and a selfsimilar mound that arises when the boundary flux vanishes. These generalized solutions provide practical benchmarks for evaluating numerical models and understanding how nonlinear hydraulic properties influence the evolution of groundwater flow in unconfined aquifers.</p>