The slow inward flow of the hot gas in elliptical galaxy cooling flows is nearly impossible to detect directly due to instrumental limitations. However, in rotating galaxies, if the inflowing gas conserves angular momentum, it will eventually form a disk. The X-ray signature of this phenomenon is a flattening of the X-ray isophotes in the inner 1-10 kpc region. This effect is observable, so we have searched for it in X-ray observations of six rotating and non-rotating early-type galaxies, obtained mainly with the ROSAT PSPC and HRI imagers. The ellipticities of the X-ray emission never increase toward the central region, nor are the X-ray ellipticities significantly greater than the ellipticities for the optical stellar emission. Central ellipticities in excess of 0.5 were expected in rotating ellipticals whereas values of 0-0.2 are measured. The failure to detect the expected signature requires a modification to the standard cooling flow picture, possibly including partial galactic winds, rapid mass drop-out, or turbulent redistribution of angular momentum
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