Is the dark matter halo of the Milky Way flattened?

We performed an extended analysis of the parameter space for the interaction of the Magellanic System with the Milky Way ( MW). The varied parameters cover the phase space parameters, the masses, the structure, and the orientation of both Magellanic Clouds, as well as the flattening of the dark matter halo of the MW. The analysis was done by a specially adopted optimization code searching for the best match between numerical models and the detailed HI map of the Magellanic System by Bruns et al. (2005, A&A, 432, 45). The applied search algorithm is a genetic algorithm combined with a code based on the fast, but approximative restricted N-body method. By this, we were able to analyze more than 10(6) models, which makes this study one of the most extended ones for the Magellanic System. Here we focus on the flattening q of the axially symmetric MW dark matter halo potential, that is studied within the range 0.74 <= q <= 1.20. We show that creation of a trailing tail (Magellanic Stream) and a leading stream (Leading Arm) is quite a common feature of the Magellanic System-MW interaction, and such structures were modeled across the entire range of halo flattening values. However, important differences exist between the models, concerning density distribution and kinematics of HI, and also the dynamical evolution of the Magellanic System. Detailed analysis of the overall agreement between modeled and observed distribution of neutral hydrogen shows that the models assuming an oblate (q < 1.0) dark matter halo of the Galaxy allow for better satisfaction of HI observations than models with other halo configurations.