The pterygopalatine part of the maxillary artery is located within the pterygopalatine fossa, going anterior to the lateral pterygoid muscle. It provides several branches that accompany similarly named branches of the maxillary nerve (CN V2).

The branches of the pterygopalatine part of the maxillary artery are as follows:

Posterior superior alveolar artery - it reenters the infratemporal fossa via the pterygomaxillary fissure and via the alveolar foramina located on the maxillary tuberosity reaches the alveolar canals where it gives dental branches to the posterior upper teeth and gingival branches to the gingiva of the same area;
Infraorbital artery - it reaches the orbit via the inferior orbital fissure and enters the infraorbital sulcus and canal, then the artery emerges on the anterior surface of the face via the infraorbital foramen; after leaving the foramen, this artery divides into smaller terminal branches that supply the muscles of the facial expression and the skin from the orbital rim to the oral rim; in the orbit, the infraorbital artery gives branches to the muscles located on the inferior wall of the orbit; when going through the infraorbital canal this artery gives anterior and middle superior alveolar arteries which enter into the maxilla and give even smaller dental and gingival branches to the upper anterior and middle teeth and gingiva, and also branches to the mucous membrane of the maxillary sinus;
Sphenopalatine artery - this artery reaches the nasal cavity via the sphenopalatine foramen and supplies the mucous membrane of the sphenoid sinus and mucous membrane of the lower and posterior aspects of the nasal cavity; the terminal branch of the sphenopalatine artery goes through the incisive canal and supplies the mucous membrane of the anterior part of the hard palate, as well as it forms an anastomosis with the descending palatine artery;
Descending palatine artery - this artery reaches the inferior surface of the palate via the greater palatine canal; gives branches on the surface of the hard palate and in the muscles of the soft palate and mucous membrane of the soft palate;
Pharyngeal artery - supplies the pharynx;
Artery of the pterygoid canal - goes through the pterygoid canal and supplies the auditory tube and mucous membrane of the nasopharynx.

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Pterygopalatine part of maxillary artery