1. Yoga-Hayagrīva, seated in padmāsana, holding in the upper arms discus and conch and in the lower ones the Vedas (represented as a thin book) and the jñānamudrā
2. Lakṣmī-Hayagrīva, seated in padmāsana and with the same attributes, but with Lakṣmī sitting on his left knee

Both are very well-spread, in temples, paper and cloth paintings and both appear to originate from Veṅkaṭanātha’s writings, the Yoga-Hayagrīva from his Hayagrīvastotra whereas the Lakṣmī-Hayagrīva seems to be a modification of the first according to the theology of Śrī Vaiṣṇavism (in which Lakṣmī is inseparable from Viṣṇu) and is described in Veṅkaṭanātha’s Śatadūṣaṇī. The center of his cult seems to have been the temple at Thiruvahindrapuram, which is also considered the “home-temple” of the Vaṭakalai religion. Unfortunately, I have never been there, nor can I plan a trip there in the immediate future. Thus, I would welcome comments and corrections by the readers.

The following one is a photo of the whole temple.

Next follows the photo of (possibly, since I have not been there) the main icon of Hayagrīva found either in the main temple or in one nearby. Readers will immediately recognise that it conforms to the Yoga-Hayagrīva typology.

Last, an icon of Lakṣmī-Hayagrīva, whose origin I do not know. Unfortunately, I have not been able to find a good image of the Lakṣmī-Hayagrīva image in the Thiruvahindrapuram temple (it is reproduced as Fig. 12a in Sridhara Babu 1990).
For more information on Hayagrīva and his connection with Veṅkaṭanātha, you can see this presentation. On Hayagrīva in general, see the posts under this tag.