Your question seems to be a bit unclear. (0, 0) is not in the solution set. In fact neither of your given curves passes through this point. As to the rest we have

and

Since r = r this means

so

You have mentioned the only two correct solutions. ( pi / 3 is the reference angle so there is another solution 4 pi / 3, or -pi / 3 which you already gave.

-Dan