There is no prerequisites for eligibility in a GMC or a Government Quota seat in a Private Medical College. So it is not clear what you mean when you say you are not eligible for a Government Seat.  in India we have Government Medical Colleges, Private Medical Colleges, Government Controlled Self financing Medical Colleges, Private Self Financing Medical Colleges and Deemed/Central Universities. And admission to all these Institutions is conducted through NEET score. The admissions are conducted at All India and State Counselling levels. Under All India Quota, All the 15% of All India Quota Seats except for Jammu and Kashmir, 100% Seats at Deemed and Central Universities, IP Quota seats in ESIC Colleges, Seats at AFMC, AIIMS and JIPMER are filled by DGHS. The 85% State Quota and Private Colleges are filled by the respective State Authorities. With 500 marks you can participate in All India Counselling and you can get a seat in Deemed/Central Universities. If you belong to Reserved Categories, you can also get a seat in a GMC. At State level, you can participate in State Counselling and you will get a seat atleast in Government Quota of Private Medical College depending on your State. In few States like Tamilnadu, Karnataka, you might even get a GMC seat. For a seat in any College, you need to participate in the respective Counselling.