I also wanted to give both the examinations and I have sent an mail to the Punjab university's official website regarding this issue if they won't response then i'm going to give CLAT for sure and will go to Patiala for pursuing my Integrated LLB degree... You and me have common interests.

Hi,

CLAT is a national level exam. It is conducted for admission to programs offered by the 22 participating NLUs.  PU LLB is a entrance exam for admission to pujab University. The  Entrance exam dates for both the exam are same i.e. 24th may

You can mail to PU LLB entrance authority at uglaw@pu.ac.in about your problem.You must mention your name, application no. and other information related to your problem in the email.

Hope they resolve your problem. If they not hear your problem then you have to give that exam which is your higher priority or deciding institution where you want to take admission.

All the best.