Please see my comment to the comment to the question by PIEBALDconsult. Picture this:
p1 p2
\ /
r \ / r
\ /
\/
center
This is how the location of the center looks in general case, when you have the continuum of solutions. Apparently, the center of the sphere of radius r lies on the sphere surrounding each of the points, p1 as a center, or p2 as a center. The two spheres intersect forming a circle (perpendicular to the plane of my drawing, with the center which lies symmetrically between p1 and p2). Each point of this circle is a solution.
If the distance between p1 and p2 is exactly 2*r, you have only one solution:
p1 p2
------------+------------
center
r r
Finally, if the distance between p1 and p2 is greater that 2*r, there are no solutions.
This is an elementary problem on high-school geometry, in fact, off-topic.
—SA