Equivariant embeddings of low dimensional symmetric planes

We determine the class of all locally compact stable planesM of positive dimensiond4 which admit a reflection at each point of some open setUM. Apart from the expected possibilities (planes defined by real and complex hermitian forms, and almost projective translation planes), one obtains (subplanes of)H. Salzmann's modified real hyperbolic planes [14; 5.3] and one exceptional plane which was not known before. The caseU=M has been treated [9] and is reproved here in a simpler way. The solution to the problem indicated in the title constitutes the main step in the proof of our results.