Given a single algebraic input-output equation, we present a method for
finding different representations of the associated system in the form of
rational realizations; these are dynamical systems with rational right-hand
sides. It has been shown that in the case where the input-output equation is of
order one, rational realizations can be computed, if they exist. In this work,
we focus first on the existence and actual computation of the so-called
observable rational realizations, and secondly on rational realizations with
real coefficients. The study of observable realizations allows to find every
rational realization of a given first order input-output equation, and the
necessary field extensions in this process. We show that for first order
input-output equations the existence of a rational realization is equivalent to
the existence of an observable rational realization. Moreover, we give a
criterion to decide the existence of real rational realizations. The
computation of observable and real realizations of first order input-output
equations is fully algorithmic. We also present partial results for the case of
higher order input-output equations