  It is shown that if It is shown that if
  \begin{equation}\label{abstract_eq}
  f(z+1)^n=R(z,f),\tag{\dag}
  \end{equation} where $R(z,f)$ is rational in $f$ with meromorphic
coefficients and $\deg_f(R(z,f))=n$, has an admissible meromorphic solution,
then either $f$ satisfies a difference linear or Riccati equation with
meromorphic coefficients, or \eqref{abstract_eq} can be transformed into one in
a list of ten equations with certain meromorphic or algebroid coefficients. In
particular, if \eqref{abstract_eq}, where the assumption $\deg_f(R(z,f))=n$ has
been discarded, has rational coefficients and a transcendental meromorphic
solution $f$ of hyper-order $<1$, then either $f$ satisfies a difference linear
or Riccati equation with rational coefficients, or \eqref{abstract_eq} can be
transformed into one in a list of five equations which consists of four
difference Fermat equations and one equation which is a special case of the
symmetric QRT map. Solutions to all of these equations are presented in terms
of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic
functions which are solutions to a difference Riccati equation. This provides a
natural difference analogue of Steinmetz' generalization of Malmquist's
theorem.
