Classes of polynomials preserving generalized pointlike partitions of their infinite domain

First-degree polynomials over rings $A=\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ are considered. Closed classes of polynomials preserving partitions of the domain $A$ into a single infinite subset and finite number of finite ones are analysed. Contents of such classes is determined. As well it is proved that recognition of preserving these partitions by arbitrary-degree polynomials ower ring $\mathbb{Z}$ is algorithmically unsolvable.