  The genuine Kaluza-Klein-like theories (with no fields in addition to gravity
with torsion) have difficulties with the existence of massless spinors after
the compactification of some of dimensions of space\cite{witten}. We
demonstrate in this letter on an example of a flat torus - as a compactified
part of an (1+5)-dimensional space - that for constant fields and appropriate
boundary conditions there exists in the 1+3 dimensional space a massless
solution, which is mass protected and chirally coupled with a Kaluza-Klein
charge to the corresponding gauge field.
