Dear Sangeetha,
In the CMM level 5 company where you applied had done background check on all your credentials and found your past employer company name to be listed under the "fake companies" category in their database, so they have to abide by their hiring policies. There is no HR/Reference check will help here until and unless you have solid evidences to prove their listing to be wrong.
You should first try to find out the reason behind calling your past employer company to be fake and as Tajsateesh advised, do consult your previous company HR Head on this issue. This will indeed prepare you further applying in other companies as well.
As a employee background screening company we can help you verify your past employer background history.
Best,
Employee background verification | 360 degree background check
You may also wish to raise this question on our LinkedIn group
Stern Advisory India - Employee Background Screening Intelligence | LinkedIn