  A very general class of axially-symmetric metrics in general relativity (GR)
that includes rotations is used to discuss the dynamics of
rotationally-supported galaxies. The exact vacuum solutions of the Einstein
equations for this extended Weyl class of metrics allow us to deduce rigorously
the following: (i) GR rotational velocity always exceeds the Newtonian velocity
(thanks to Lenz's law in GR); (ii) A non-vanishing intrinsic angular momentum
($J$) for a galaxy demands the asymptotic constancy of the Weyl (vectorial)
length parameter ($a$) -a behavior identical to that found for the Kerr metric;
(iii) Asymptotic constancy of the same parameter $a$ also demands a plateau in
the rotational velocity. Unlike the Kerr metric, the extended Weyl metric can
and has been continued within the galaxy and it has been shown under what
conditions Gau\ss\ \&\ Amp\'ere laws emerge along with Ludwig's extended GEM
theory with its attendant non-linear rate equations for the velocity field.
Better estimates (than that from the Newtonian theory) for the escape velocity
of the Sun and a reasonable rotation curve \&\ $J$ for our own galaxy has been
presented.
