Most Kähler manifolds do not contain any submanifolds that are simultaneously minimal and Lagrangian since the combination of the two conditions is equivalent to an overdetermined system of PDE for the submanifold that is generally incompatible.
However, in case the 2n-manifold M is Kähler-Einstein, the situation is different. I prove that, in this case, the overdetermined system is involutive. In fact, every real-analytic submanifold of dimension n-1 that is sub-Lagrangian (i.e., on which the Kähler form vanishes) lies in a circle of n-manifolds, each of which is minimal and Lagrangian (these will not generally be compact).
Quite recently, these minimal Lagrangian manifolds have become a subject of interest to physicists (in the physics literature this comes under the heading of `BPS states' in string theory). Works in this area in physics can be found by such authors as Vafa, Witten, Yau, and Zaslow. On the mathematical side, R. Schoen and J. Wolfson have worked in this area, not to mention R. Harvey and H. B. Lawson (in the Ricci-flat case).
For some reason, I have been getting many requests for reprints of this article lately, perhaps because of it being referenced in the physics literature. However, I never received reprints for this article and it predates my use of TeX by many years; all I have is a typescript. Since the article has been published and is easily available, I am not making photocopies to mail out.