Minimal surfaces in $S^3$ foliated by circles

We deal with minimal surfaces in the unit sphere $S^3$, which are one-parameter families of circles. Minimal surfaces in $\R^3$ foliated by circles were first investigated by Riemann, and a hundred years later Lawson constructed examples of such surfaces in $S^3$. We prove that in $S^3$ there are only two types of minimal surfaces foliated by circles, crossing the principal lines at a constant angle. The first type surfaces are foliated by great circles, which are bisectrices of the principal lines, and we show that these minimal surfaces are the well-known examples of Lawson. The second type surfaces, which are new in the literature, are families of small circles, and the circles are principal lines. We give a constructive formula for these surfaces. An application to the theory of minimal foliated semi-symmetric hypersurfaces in $\R^4$ is given.