Separable triaxial potential-density pairs in MOND
  We study mass models that correspond to MOND (triaxial) potentials for which
the Hamilton-Jacobi equation separates in ellipsoidal coordinates. The problem
is first discussed in the simpler case of deep-MOND systems, and then
generalized to the full MOND regime. We prove that the Kuzmin property for
Newtonian gravity still holds, i.e., that the density distribution of separable
potentials is fully determined once the density profile along the minor axis is
assigned. At variance with the Newtonian case, the fact that a positive density
along the minor axis leads to a positive density everywhere remains unproven.
We also prove that (i) all regular separable models in MOND have a vanishing
density at the origin, so that they would correspond to centrally dark-matter
dominated systems in Newtonian gravity; (ii) triaxial separable potentials
regular at large radii and associated with finite total mass leads to density
distributions that at large radii are not spherical and decline as ln(r)/r^5;
(iii) when the triaxial potentials admit a genuine Frobenius expansion with
exponent 0<epsilon<1, the density distributions become spherical at large
radii, with the profile ln(r)/r^(3+2epsilon). After presenting a suite of
positive density distributions associated with MOND separable potentials, we
also consider the important family of (non-separable) triaxial potentials V_1
introduced by de Zeeuw and Pfenniger, and we show that, as already known for
Newtonian gravity, they obey the Kuzmin property also in MOND. The ordinary
differential equation relating their potential and density along the z-axis is
an Abel equation of the second kind that, in the oblate case, can be explicitly
reduced to canonical form.
