Context. Ambipolar diffusion is increasingly recognised as a key process in the lower solar atmosphere. Its highly non-linear behaviour has many non-intuitive aspects. Aims. We seek to (a) study 1D Cartesian ambipolar diffusion near null points; (b) characterise the non-linear eigenmodes for ambipolar diffusion; and (c) propose tests for ambipolar diffusion solvers in MHD codes. Methods. (a) We employed a direct analysis to obtain analytical solutions for ambipolar diffusion. (b) To study the eigenmodes, we solved the ODE for self-similar solutions of the 1D ambipolar diffusion equation using phase-plane techniques. We also solved the general time-dependent 1D problem for initial conditions of interest. (c) We tested the Bifrost code by trying to reproduce the behaviour of the eigenmodes. Results. (a) A stagnation-point flow solution was found with a uniform flux transfer rate across three regions: an external advection region; an internal ambipolar diffusion region with magnetic profile B ∝ x1/3; and an innermost Ohmic region with B ∝ x; in the latter, flux annihilation occurs at a rate imposed by the advection. (b) Both symmetric and antisymmetric eigenmode solutions to the ambipolar diffusion problem were found with sharp current sheets at the internal nulls. The time evolution of the eigenmodes (pure or perturbed) was probed, showing how higher order eigenmodes, or perturbed ones, evolve over time towards the lowest order allowable eigenmodes. (c) The Bifrost code reproduces the behaviour of the eigenmodes with excellent accuracy. Conclusions. Stagnation-point configurations exist with ambipolar diffusion carrying magnetic flux in an inner layer and serving as an intermediary between the external advection and the flux dissipation and annihilation at an Ohmic-diffusion core around the null. Our tests are compatible with the hypothesis that zero-flux higher harmonics of the self-similar equation evolve toward either the first symmetric or antisymmetric harmonic. The self-similar solutions can serve as strong tests for ambipolar diffusion solvers in general MHD codes.