  In this paper we consider homothetic Killing vectors in the class of
stationary axisymmetric vacuum (SAV) spacetimes, where the components of the
vectors are functions of the time and radial coordinates. In this case the
component of any homothetic Killing vector along the $z$ direction must be
constant. Firstly, it is shown that either the component along the radial
direction is constant or we have the proportionality $g_{\phi\phi}\propto
g_{\rho\rho}$, where $g_{\phi\phi}>0$. In both cases, complete analyses are
carried out and the general forms of the homothetic Killing vectors are
determined. The associated conformal factors are also obtained. The case of
vanishing twist in the metric, i.e., $\omega= 0$ is considered and the complete
forms of the homothetic Killing vectors are determined, as well as the
associated conformal factors.
