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Einstein and ROSAT observations of Q0957+561 indicate that X-ray variations of the fluxes of images A and B do not follow those in the optical band. Accounting for a 1.5 year difference in the signal propagation time for the two images, we find the X-ray flux ratio to be 2.2 $\pm$ 0.3 in 1991 - 1992 which differs significantly from the corresponding``$phased$'' values of 0.75 for the radio and broadline region emission and 1.05 for the optical continuum. This observed wavelength dependence of the phased flux ratios may imply that one or both images are being microlensed by stars or clumps of dark matter. This scenario is consistent with the pattern of the differences in emission radii in these wavelength bands, as determined from variability arguments. However it is also possible that the adopted time delay of 1.5 years is not correct. Large X-ray variations that might occur on the timescale of the error of the time delay may significantly affect the phased flux ratios. By taking advantage of the magnification of the quasar images and of the quasar environment we detected a possibly thermal emission component in the ROSAT PSPC spectrum of 0957+561 with an estimated temperature of about 0.4 keV and a flux of about 5 x 10$^{-13}$ ergs sec$^{-1}$ cm$^{-2}$. This component may corresponds to the arc like structure that is spatially resolved in the Einstein HRI observation of 0957+561 and that may originate from a cooling flow centered at about 7 kpc from the quasar. From the measured bound on the ionization state, NHI/NH(intrinsic) $>$ 0.07 (3$\sigma$), we computed an upper bound on the spatial extent of the damped Lyman-$\alpha$ absorber of 30 h$^{-1}_{50}$ kpc. We placed as a lower bound on the spatial extent of the HI cloud,the comoving linear separation of 0.2 h$^{-1}_{50}$ kpc between light beams A and B at the redshift of the absorber.
