On some questions around Berest's conjecture

Abstract: Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove the following two results. Assume there exists a non-zero polynomial $f(X,Y)\in K[X,Y]$, which has a non-trivial solution $(P,Q)\in A_{1}^{2}$ with $[P,Q]=0$, and the number of orbits under the group action of $Aut(A_1)$ on solutions of $f$ in $A_{1}^{2}$ is finite. Then the Dixmier conjecture holds, i.e $\forall \varphi\in End(A_{1})-{0}$, $\varphi$ is an automorphism. Assume $\varphi$ is an endomorphism of monomial type (in particular, it is not an automorphism, see theorem 4.1). Then it has no non-trivial fixed point, i.e. there are no $P\in A_1$, $P\notin K$, s.t. $\varphi (P)=P$.