Some galaxies possess a polar ring, most of which are of type SO. The most widely accepted explanation for the formation of polar ring galaxies is that accreted gas settles onto orbits that are more frequently contained either within the equatorial plane or in polar planes. As Polar Ring Galaxies are typically SO galaxies, and the ring is gaseous, blue and undergoing star formation, their nature does not greatly differ from gas-rich SO galaxies, commented above, but the importance of Polar Ring Galaxies with respect to the problem of dark matter halos is that information about the overall gravitational potential can be obtained in two planes: the disk and the ring planes, thus potentially constraining the shape of the halo.
Sackett (1999)
has reviewed the topic of the shape of halos, with
Polar Ring Galaxies being one of the most interesting techniques to
determine it. Assuming
it to be a triaxial ellipsoid, the ovalness (b/a) and the flattening
(c/a), where
a
bc, are to be determined. Intrinsic ovalness
of the density distribution in the disk can be used to trace the
non-axisymmetry of the halo in any face-on spiral galaxy
(Rix and Zaritsky, 1995)
finding a value for
b/a 
0.85. This figure
together with those obtained by other methods summarized by
Sackett (1999)
indicates that b/a > 0.7, so that the
unobserved halo could have a higher axisymmetry.
However, the flatness is more difficult to assess, and polar ring galaxies are specially suitable for this purpose. This point is particularly important because it can provide information about the baryonic and dissipative nature of the halo dark matter (Pfenniger, Combes and Martinet, 1994).
Polar rings have very large radii, of about 20 stellar disk radial scale lengths, and therefore the perturbing influences of central luminous components are less important, and observations would provide the flatness c/a of the dark halo. This task has been carried out by several authors since the pioneering work by Schweizer, Whitmore and Rubin (1983) including more recent analyses by Combes and Arnaboldi (1996) and others (see the review by Sackett, 1999, and references therein).
From the analysis of polar ring galaxies, Sackett concludes that halos
are highly flattened,
0.3
c/a0.6, which coincides with a similar
conclusion from the flattening of the X-ray halos of elliptical
galaxies
(Buote and Canizares, 1998).
Observations of gravitational lensing
(Kochanek, 1995)
also suggest greatly flattened halos.
Other methods to determine c/a not based on PRG have been reported. The conclusions are model dependent and, in some cases, are even based on hypotheses that are not completely demonstrated. For instance, the analysis of warps (New at al, 1998) and of flaring (Olling and Merrifield, 1997; Becquaert and Combes, 1998) are based on interesting models, but which are not free of alternative explanations.
Ashman (1992) points out that polar ring galaxies are unusual objects and therefore their hypothetical halos may be atypical. For instance, the merging process from which they have originated could have given rise to a flattened halo. Alternatively, the settling of a polar ring in the accretion process may require a flattened halo, in which case, the scarcity of polar ring galaxies would suggest that most halos are spherical. The influence of magnetic fields on the dynamics of these rings may be non ignorable.
As the polar ring contains HI, it is useful to detect dark matter, but as the host galaxy is lenticular and usually gas-poor, we cannot benefit from the standard analysis of spirals. Therefore the study of the exceptional galaxy NGC 660, the only polar-ring spiral galaxy known is very important. It has been extensively studied by van Driel et al. (1995) and van Driel and Combes (1997). The disk has a flat rotation curve and the polar ring is a rising one, which is rather puzzling. No conclusion about the flatness was reached, although the authors noted that several problems cannot be ignored, such as the fact that the ring is very massive, so that it cannot be considered to be formed by test particles tracing the potential, together with the fact that, obviously, the polar ring velocity and the disk velocity cannot be measured at the same radius. These objections raised by van Driel and Combes (1997) also hold for other dark matter studies in polar ring galaxies.