On the maximal dual pairs of invariant subspaces of $J$-self-adjoint operators

In the $J$-spaces $\mathfrak{H}=\mathfrak{H}_1\oplus\mathfrak{H}_2$, with the infinite-dimensional components $\mathfrak{H}_k=P_k\mathfrak{H}$ ($k=1,2$), we can always find an operator $A$, for which there are at least two distinct invariant maximal dual pairs, such that if $[x,x]=0$ and $[Ax,x]=0$, then $x=0$.