The q-division ring and its fixed rings

We prove that the fixed ring of the q-division ring kq(x,y) under any finite group of monomial automorphisms is isomorphic to kq(x,y) for the same q. In a similar manner, we also show that this phenomenon extends to an automorphism that is defined only on kq(x,y) and does not restrict to kq[x±1,y±1]. We then use these results to answer several questions posed by Artamonov and Cohn about the endomorphisms and automorphisms of kq(x,y).