jmc

There are two cases, one where the $AB$ is the base, and the other where $AB$ is a leg.

For the case where $AB$ is the base, we can create the third point $C$ anywhere on the line perpendicular to $AB$ at the midpoint of $AB$. There are $4$ points on that line.

For the case where $AB$ is a leg, since $AB$ is two units, we can create a point $C$ two units away from either $A$ or $B$. There are two such points.

In total, there are $2+4=\boxed{6}$.