On injective endomorphisms of the semigroup ${\mathit{B}}_{\mathbb{Z}}^{{\mathcal{F}}^2}$ with the two-element family ${\mathcal{F}}^2$ of inductive nonempty subsets of $\omega$

We describe injective endomorphisms of the semigroup ${\mathit{B}}_Z^{{\mathcal{F}}^2}$ with the two-element family ${\mathcal{F}}^2$ of inductive nonempty subsets of $\omega$. In particular we show that every injective endomorphism $\mathfrak{e}$ of ${\mathit{B}}_Z^{{\mathcal{F}}^2}$ is presented in the form $\mathfrak{e}={\mathfrak{e}}_0\mathfrak{a}$, where ${\mathfrak{e}}_0$ is an injective $(0,0,[0))$-endomorphism of ${\mathit{B}}_{\mathbb{Z}}^{{\mathcal{F}}^2}$ and $\mathfrak{a}$ is an automorphism $\mathfrak{a}$ of ${\mathit{B}}_{\mathbb{Z}}^{{\mathcal{F}}^2}$. Also we describe all injective $(0,0,[0))$-endomorphisms ${\mathfrak{e}}_0$ of ${\mathit{B}}_{\mathbb{Z}}^{{\mathcal{F}}^2}$, i.e., such that $(0,0,[0)){\mathfrak{e}}_0=(0,0,[0))$.