We present a set of highly flattened galaxy models with asymptotically constant rotation curves .
The mass density in the equatorial plane falls like ( distance ) ^ { -1 } at large radii .
Although the inner equidensity contours may be spherical , oblate or prolate , the outer parts are always severely flattened .
The elongated shape is supported by rotation or tangential velocity anisotropy .
The models are thickened Mestel discs , and form a previously undiscovered part of the Miyamoto & Nagai sequence of flattened galaxies .
The properties of the models – axis ratios , velocity dispersions , streaming velocities and distribution functions – are all discussed in some detail .

We pose the question : are extremely flattened or disk-like haloes possible for the Milky Way galaxy ?
This has never been examined before , as very flattened halo models were not available .
We fit the rotation curve and the vertical kinematics of disc stars in the solar neighbourhood to constrain the overall shape of the Galaxy .
Denoting the ratio of polar axis to major axis by q , we show that models with q \lesssim 0.57 can not simultaneously reproduce the in-plane and out-of-plane constraints .
The kinematics of the Sagittarius galaxy also strongly disfavour models with high flattening , as the orbital plane precession is too great and the height reached above the Galactic plane is too small .
At least for our Galaxy , the dark halo can not be flatter than E4 ( or axis ratio q \sim 0.57 ) at the Solar circle .
Models in which the dark matter is accounted for by a massive baryonic disc or by decaying neutrinos are therefore ruled out by constraints from the rotation curve and the vertical kinematics .