How to check the integrity of a key received through mail using symmetric encryption?
Unfortunately, symmetric encryption do not provide integrity and digital signature functions. This is because symmetric encryption uses the same key for exncryption and decryption. This means that any who owns the decryption key can alter the data and no one will notice that the data is altered, because the same key was used for encryption.

This means that data integrity can be provided only by using asymmetric cryptography.