The flattening rotation velocity $v(r)\to {\rm constant}$ found by Vera Rubin and collaborators and very apparent in the SPARC galaxy-rotation data coincides with Kepler's law in one less dimension. Thus, it is naturally reproduced by elongated dark matter distributions with the axis of prolateness perpendicular to the galactic plane. This theoretical understanding is borne out by the detailed fits to the rotation data that we here report: for equal dark matter profile, elongated distributions provide smaller $χ^2$ than purely spherical ones. We also propose to use the geometric mean of the individual halo ellipticities, as opposed to their arithmetic average, because $s=c/a\in (0,\infty)$ corresponds to spherical haloes for $s=1$, so that the usually reported average is skewed towards oblateness and fails to reveal the large majority of prolate haloes. Several independently coded fitting exercises concur in yielding $s<1$ for most of the database entries and the oblate exceptions are understood and classified. This likely prolateness is of consequence for the estimated dark matter density near Earth.

26 pages, original primary work, about 40 plots, 1.4 Mbytes. v2: small latex and edition fixes (repeated label, overlapping paragraphs, shortened a table to 1-column format, etc.)