Local rigidity and nonrigidity of symplectic pairs

It is shown that indefinite strictly almost K?hler and opposite K?hler structures (J, J′) on a four-dimensional manifold with J-invariant Ricci operator are rigid, thus extending a previous result of Apostolov, Armstrong and Drăghici from the positive definite case to the indefinite one. In contrast to this, examples of nonhomogeneous four-dimensional manifolds which admit strictly almost paraK?hler and opposite paraK?hler structures (\mathfrakJ,\mathfrakJ^￠){(\mathfrak{J},\mathfrak{J}^{\prime})} with \mathfrakJ{\mathfrak{J}} -invariant Ricci operator are shown.