Solution

There are two scenarios in which the image of the source coincides with the source.

1. Light rays (Red) emerging from the lens converge to the center of the sphere, in which case they are reflected and return on the same path. In this case, the position of the object in front of the lens is

$$u=\frac{df}{d-f},$$

which is possible if $d>f$.

2. Light rays (Blue) emerging from the lens converge to the front of the sphere, in which case they are reflected and return on the path symmetric to the original path with respect to the principal axis. In this case, the position of the object in front of the lens is

$$u=\frac{(d-R)f}{(d-R)-f},$$

which is possible if $d-R>f$.