Can a CPT violating ether solve all electron (anti)neutrino puzzles?

Assuming that CPT is violated in the neutrino sector seems to be a viable alternative to sterile neutrinos when it comes to reconciling the LSND anomaly with the remainder of the neutrino data. There are different (distinguishable) ways of incorporating CPT violation into the standard model, including postulating m ≠ m̄. Here, I investigate the possibility of introducing CPT violation via Lorentz-invariance violating effective operators ("ether" potentials) which modify neutrino oscillation patterns such as ordinary matter effects. I argue that, within a simplified two-flavor-like oscillation analysis and restricting the analysis to the lowest order Lorentz-invariance violating operators only, one cannot solve the solar neutrino puzzle and LSND anomaly while still respecting constraints imposed by other neutrino experiments, and comment on whether significant improvements should be expected from a three-flavor analysis. If one turns the picture upside down, some of the most severe constraints on such CPT violating terms can already be obtained from the current neutrino data, while much more severe constraints can arise from future neutrino oscillation experiments.