Additive mappings between Hermitian matrix spaces preserving rank not exceeding one

Let K be a field of characteristic not two or three with an automorphism − of order two. Let F={a∈K:a¯=a}. We characterize additive mappings T from one F-vector space of Hermitian matrices over K to another that preserve rank less than or equal to one for the following cases: (i) the image of T contains a matrix of rank at least three, (ii) every nonzero endomorphism of F is surjective.