Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation

It is proved that the quotient space of the four-dimensional quaternionic projective space by the automorphism group of the quaternionic algebra becomes the 13-dimensional sphere while quotioned by the quaternionic conjugation. This fact and its various generalizations are proved using the results of the theory of the hyperbolic partial differential equations, providing also the proof of the theorem (which was, it seems, known to L. S. Pontriagin in the 1930s) claiming that the quotient of the complex projective plane by the complex conjugation is the 4-sphere.