  We prove that the fixed ring of the $q$-division ring $k_q(x,y)$ under any
finite group of monomial automorphisms is isomorphic to $k_q(x,y)$ for the same
$q$. In a similar manner, we also show that this phenomenon extends to an
automorphism that is defined only on $k_q(x,y)$ and does not restrict to
$k_q[x^{\pm1},y^{\pm1}]$. We then use these results to answer several questions
posed by Artamonov and Cohn about the endomorphisms and automorphisms of
$k_q(x,y)$.
