Minimal real Kaehler submanifolds

We show that generic rank conditions on the second fundamental form of an isometric immersion $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ a Kaehler manifold complex dimension $n\geq 2$ into Euclidean space with low codimension $p$ implies submanifold has to be minimal. If $M^{2n}$ if simply connected, this amounts existence one-parameter associated family minimal immersions unless $f$ is holomorphic.