Ranks of least squares solutions of the matrix equation AXB=CAXB=C

For a complex matrix equation AXB=CAXB=C, we solve the following two problems: (1) the maximal and minimal ranks of least square solution XX to AXB=CAXB=C, and (2) the maximal and minimal ranks of two real matrices X0X0 and X1X1 in least square solution X=X0+iX1 to AXB=CAXB=C. We also give a necessary and sufficient condition for matrix equations AiXiBi=Ci(i=1,2) to have a common least square solution.