chicken Chickens & Pigs pig

On my farm I counted both the heads and legs of my chickens and pigs and got 70. How many of each animal are there?

Traditionally this problem is given with an additional piece of information, allowing us to write a set of linear equations. However, with just one equation in two unknowns, we get a set of possible answers.

If we allow for negative chickens and pigs, or partial animals, we would be able to graph the linear equation. However, the context of the problem only allows for a finite set of answers. Find this finite set.