  The Deficiency-One Theorem states that there exists a unique positive steady
state in each positive stoichiometric class for weakly reversible
deficiency-one mass action systems with one linkage class (regardless of the
values of the rate coefficients). The non-emptiness of the set of positive
steady states does not remain valid if we omit the weak reversibility. A
recently published paper provided an equivalent condition to the existence of a
positive steady state for deficiency-one mass action systems that are not
weakly reversible, but still has only one linkage class. Based on that result,
we characterise in this paper those of these mass action systems for which the
non-emptiness of the set of positive steady states holds regardless of the
values of the rate coefficients. Also, we provide an equivalent condition to
the existence of rate coefficients such that the set of positive steady states
is nonempty for the resulting mass action system.
