  Motivated by the study of invariant rings of finite groups on the first Weyl
algebras $A_{1}$ (\cite{AHV}) and finding interesting families of new
noetherian rings, a class of algebras similar to $U(sl_{2})$ were introduced
and studied by Smith in \cite{S}. Since the introduction of these algebras,
research efforts have been focused on understanding their weight modules, and
many important results were already obtained in \cite{S} and \cite{Ku}. But it
seems that not much has been done on the part of nonweight modules. In this
note, we generalize Kostant's results in \cite{K} on the Whittaker model for
the universal enveloping algebras $U(\frak g)$ of finite dimensional semisimple
Lie algebras $\frak g$ to Smith's algebras. As a result, a complete
classification of irreducible Whittaker modules (which are definitely infinite
dimensional) for Smith's algebras is obtained, and the submodule structure of
any Whittaker module is also explicitly described.
