  In this paper, we consider certain linear-fractional branching processes with
immigration in varying environments. For $n\ge0,$ let $Z_n$ counts the number
of individuals of the $n$-th generation, which excludes the immigrant which
enters into the system at time $n.$ We call $n$ a regeneration time if $Z_n=0.$
We give first a criterion for the finiteness or infiniteness of the number of
regeneration times. Then, we construct some concrete examples to exhibit the
strange phenomena caused by the so-called varying environments. It may happen
that the process is extinct but there are only finitely many regeneration
times. Also, when there are infinitely many regeneration times, we show that
for each $\varepsilon>0,$ the number of regeneration times in $[0,n]$ is no
more than $(\log n)^{1+\varepsilon}$ as $n\rightarrow\infty.$
