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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 Gauß \&\ Ampére 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.