From the mid 1980's, groups at CSIRO (including the author), Griffith U. and U. of Arizona (including A. Warrick) adapted integrable nonlinear convection-diffusion equations to obtain realistic one-dimensional solutions for transient unsaturated flow in soil. The solution with constant-flux boundary conditions has been of most interest but the solution with constant-concentration boundary conditions has so far defied our best efforts. This problem can be transformed to the standard Stefan problem for solidification, with latent heat release, linear heat conduction and additional steady heat extraction occurring at the free boundary. The standard scale-invariant Neumann solution is the leading term of the early-time solution, which neglects the steady heat extraction. If we choose independent coordinates to be canonical coordinates of the scaling symmetry, then separation of variables is admissible at all levels of correction for the non-invariant problem. The full solution is a power series in t^1/2 for which remarkably, each term satisfies the governing equation.