K3 surfaces constitute a remarkable family of complex surfaces. Examples can be built by standard geometric constructions in complex geometry or algebraic geometry. Here I will describe a beautiful (and more original) construction due to Koike and Uehara, where such surfaces are obtained by gluing along a real hypersurfaces, which is said to be linear Levi-flat. This leads to the question of which K3 surfaces contain such linear Levi-flat hypersurfaces. This question might seem artificial, but we will see that we can give a very partial answer by using the construction by Koike and Uehara and beautiful ideas of Verbitsky. We will thus obtain an existence result by using ingredients from complex geometry, algebraic geometry, dynamical systems, ergodic theory and homogeneous spaces for Lie groups.