Partially Homomorphic Cryptographic Schemes - Deterministic vs Probabilistic
Yes, there exist other deterministic PHE schemes. Actually, you can turn any probabilistic PHE into a deterministic one. This is achieved by deriving the random coins for the encryption deterministically from the message to be encrypted, e.g., by computing the random coins as a hash of the message (using a suitable hash function compatible with the randomness space of the encryption scheme). 

Moreover, the determinism of the PHE is not related to whether the operation on plain- and ciphertexts are identical (a simple counterexample is ElGamal encryption, which is a probabilistic PHE, and uses the same operations on plaintexts and ciphertext components).