Public-key encryption with associated data
There are no schemes that I'm aware of that are specifically designed for this task, i.e. where this was advertised as feature in a description I've come across.

The most standard solution I can provide is McBits (PDF). McBits is based on XSalsa20-Poly1305 and thereby can natively supports the associated data interface if enabled by the implementation. However (good) implementations of McBits are rare, although they shouldn't be too hard given a (standard) McEliece implementation.

What you can also do, is take your generic (IND-CCA2 secure) public key encryption scheme (e.g. RSA-OAEP), encrypt a symmetric key with it and use that key for the AEAD scheme of your choice (e.g. AES-GCM) to encrypt and authenticate bulk data and to authenticate header data.

The more distinguished solution is to just take the KEM of your choice (like the one used by ECIES, DHIES or RSA-KEM), use that to derive the key and then continue normally with your AEAD scheme of choice (e.g. AES-GCM). McBits basically is this with the KEM being standard McEliece encryption of a random vector.