  We analyze the Miyamoto--Nagai substitution, which was introduced over forty
years ago to build models of thick disks and flattened elliptical galaxies.
Through it, any spherical potential can be converted to an axisymmetric
potential via the replacement of spherical polar $r^2$ with $R^2 + ( a +
\!\sqrt{z^2+b^2} )^2$, where ($R,z$) are cylindrical coordinates and $a$ and
$b$ are constants. We show that if the spherical potential has everywhere
positive density, and satisfies some straightforward constraints, then the
transformed model also corresponds to positive density everywhere. This is in
sharp contradistinction to substitutions like $r^2 \rightarrow R^2 + z^2/q^2$,
which leads to simple potentials but can give negative densities. We use the
Miyamoto--Nagai substitution to generate a number of new flattened models with
analytic potential--density pairs. These include (i) a flattened model with an
asymptotically flat rotation curve, which (unlike Binney's logarithmic model)
is always non-negative for a wide-range of axis ratios, (ii) flattened
generalizations of the hypervirial models which include Satoh's disk as a
limiting case and (iii) a flattened analogue of the Navarro--Frenk--White halo
which has the cosmologically interesting density fall-off of (distance)$^{-3}$.
Finally, we discuss properties of the prolate and triaxial generalizations of
the Miyamoto-Nagai substitution.
