Since the topic is about Falconry I will put a list of raptor birds also known as Falconiform group which includes eagles, falcons, hawks and others https://www.britannica.com/animal/falconiform
Latin names of these birds with descriptions and pictures that live in Belarus : https://ru.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D1%81%D0%BE%D0%BA_%D0%BF%D1%82%D0%B8%D1%86_%D0%91%D0%B5%D0%BB%D0%BE%D1%80%D1%83%D1%81%D1%81%D0%B8%D0%B8#%D0%A1%D0%BE%D0%BA%D0%BE%D0%BB%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BD%D1%8B%D0%B5
27 types of such birds in total. Certain birds are rare and endangered; they are protected (put in the red book as we say) by state.
If I was going to choose a bird of prey for hunting it would the northern gashawk. The common Slavic name for the bird is ястреб/jastreb with some variations. It’s around the size of a large falcon . It’s well suited for hunting in woodlands. Here’s an example gashawks hunt for pheasants in Ireland’s woodland.