The focus of this work is on a conjecture of Blaine Lawson that the only embedded minimal 2-torus in the standard 3-sphere is the Clifford torus. Our approach to resolving this conjecture is to view the larger class of minimal immersions of the plane in terms of the nullity of the immersions, the kernel of the second variation of the area functional. We examine the case where the nullity is restricted. This restriction, conveniently, allows us to construct a dimension-2 foliation (for each set of initial conditions) on $SO(4)xR^2$, the leaves of which project to minimal immersions of the plane into the sphere. We proceed to show that all compact images, all minimal immersions of the torus with restricted nullity, are known examples, the only ebedded one of which is the Clifford torus. Thus any counter-example to this conjecture would have to have large nullity. Joint work with Oscar Perdomo (Universidad del Valle/Central CT State U.)