We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear
equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides
a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with
much more general transition structures to be accommodated. One application is to give a new proof of an important
result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove
new results for downwardly skip-free chains, such as the Markov branching process and several of its many
generalizations. This permits us to establish uniqueness criteria for several models, including the general birth,
death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being
neither upwardly skip-free nor downwardly skip-free.
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