We study a strongly nonlinear equation describing the long-wave asymptotic behaviour of rapid directional solidification with weak convective coupling. Taking into account kinetic and disequilibrium effects in the basic equation, we are able to identify various nontrivial integrable subproblems that are absent in previous equations derived for this asymtotic regime. Applying then the averaging method to obtain an approximate description of solutions to the full equation, we are led to a new way of looking at wavelength selection problems, revealing a common framework for temporal and spatial processes. The method allows to establish quantitative relations between the drift velocity of parity-breaking solutions and their wavelength. Furthermore, it suggests a new wavelength-selection mechanism for equations without parity symmetry under conditions where a lateral drift is suppressed.