Calibrating data containing XX and YY is somewhat simpler. You have insufficient information to solve for instrumental polarisation and the calibrator Q and U . You have no choice but to assume that these are zero. This means that we can skip a number of steps in the calibration process.
All these assumptions will result in some error in the calibration -- this is fundamentally unavoidable if only XX and YY are measured. Provided the secondary calibrator is weakly polarised, the error is unlikely to be too substantial.