Most of the stationary first-order autoregressive integer-valued (INAR(1)) models were developed for a given thinning operator using either the forward approach or the backward approach. In the forward approach the marginal distribution of the time series is specified and an appropriate distribution for the innovation sequence is sought. Whereas in the backward setting, the roles are reversed. The common distribution of the innovation sequence is specified and the distributional properties of the marginal distribution of the time series are studied. In this article we focus on the backward approach in presence of the Binomial thinning operator. We establish a number of theoretical results which we proceed to use to develop stationary INAR(1) models with finite mean. We illustrate our results by presenting some new INAR(1) models that show underdispersion.