We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset X = {0, 1, 2,...}, in which individuals of type i may give birth to offspring of type j <= i + 1 only. For this class of processes, we study the set S of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q and whose maximum is the partial extinction probability vector (q) over tilde. In the case where (q) over tilde = 1, we derive a global extinction criterion which holds under second moment conditions, and when (q) over tilde < 1 we develop necessary and sufficient conditions for q = <(q)over tilde>. We also correct a result in the literature on a sequence of finite extinction probability vectors that converge to the infinite vector (q) over tilde.