On Some Questions around Berest's Conjecture

Let $K$ be a field of characteristic zero, and let $A_1=K[x][\partial ]$ be the first Weyl algebra. In the present paper, we prove the following two results.
Assume that there exists a nonzero polynomial $f(X,Y)\in K[X,Y]$ such that (i) $f$ has a nontrivial solution $(P,Q)\in A_{1}^{2}$ with $[P,Q]=0$; (ii) the set of solutions of $f$ in $A_{1}^{2}$ splits into finitely many $\operatorname{Aut}(A_1)$-orbits under the natural actuon of the group $\operatorname{Aut}(A_1)$. Then the Dixmier conjecture holds; i.e., every $\varphi\in \operatorname{End}(A_{1})\setminus\{0\}$ is an automorphism.
Assume that $\varphi\in \operatorname{End}(A_{1})$ is an endomorphism of monomial type. (In particular, it is not an automorphism; see Theorem 4.1.) Then $\varphi$ has no nontrivial fixed points; i.e. there exists no $P\in A_1\setminus K$ such that $\varphi (P)=P$.