Is MAC process different for symmetric and asymmetric? Message Authentication Code as far as I know works on the shared secret key. Do they work in the same way. Is there no difference between symmetric and asymmetric processes. Can anyone please explain me from message integrity and authentication perspective?
A cryptographic MAC is always symmetric By Definition. 

For those interested in its security properties, one may want to know Does data authenticity always, implicitly, provide data integrity?

The tool that provides authenticity asymmetrically is known as Digital Signature. Digital Signatures are asymmetric By Definition; they guarantees the integrity, authenticity, and non-repudiation of signed messages when the correctness of the public key of the signer is guaranteed.