  This paper is concerned with the characterizations of fixed points of the
generating function of branching processes with countably infinitely many
types. We assume each particle of type $i$ can only give offspring of type
$j\geq i$, whose number only depends on $j-i$. We prove that, for these
processes, there are at least countably infinitely many fixed points of the
offspring generating function, while the extinction probability set of the
process has only $2$ elements. This phenomenon contrasts sharply with those of
finite-type branching processes. Our result takes one step forward on the
related conjecture on the fixed points of infinite-dimensional generating
functions in literature. In addition, the asymptotic behavior of the components
of fixed point is given.
