  The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled
by two parameters $d$ and $\ell$. The aim of the present work is to investigate
the lowest weight representations of CGA with $d = 1$ for any integer value of
$\ell$. First we focus on the reducibility of the Verma modules. We give a
formula for the Shapovalov determinant and it follows that the Verma module is
irreducible if $\ell = 1$ and the lowest weight is nonvanishing. We prove that
the Verma modules contain many singular vectors, i.e., they are reducible when
$\ell \neq 1$. Using the singular vectors, hierarchies of partial differential
equations defined on the group manifold are derived. The differential equations
are invariant under the kinematical transformation generated by CGA. Finally we
construct irreducible lowest weight modules obtained from the reducible Verma
modules.
