The limit behaviour of decomposable critical branching processes with two types of particles

We consider two-dimensional branching processes $\mu(t)=(\mu_1(t),\mu_2(t))$, $t\in\{0,1,\dots\}$, with the offspring generating functions
\begin{gather*} \mathbf E\{s_1^{\mu_1(1)}s_2^{\mu_2(1)}\mid\mu(0)=(1,0)\}= F_1(s_1)=s_1+(1-s_1)^{1+\alpha_1}L_1(1-s_1), \\ \mathbf E\{s_1^{\mu_1(1)}s_2^{\mu_2(1)}\mid\mu(0)=(0,1)\}= s_2+(1-s_2)^{1+\alpha_2}L_2(1-s_2)-(A+o(1))(1-s_1), \end{gather*}
where $0<\alpha_1$, $\alpha_2\le 1$ and the functions $L_1(x)$, $L_2(x)$ are slowly varying when $x\downarrow 0$. We investigate the asymptotics of
$$ \mathbf P\{\mu(t)\ne 0\mid\mu(0)=(0,1)\},\qquad t\to\infty, $$
and prove the limit theorems for the conditional distribution of the numbers of particles.