Real-analytic coordinates for smooth strictly pseudoconvex CR structures

Let $M$ be a smooth manifold, $E$ the complexified tangent bundle, $X_1,\ldots,X_k$ local sections of $E$ satisfying the integrability condition. Under which condition there exists a (local) smooth diffeomorphism transforming the vector fields $X_1,\ldots,X_k$ into real-analytic ones? We resolve this question for the case when the vector fields generate a hypersurface type strongly pseudoconvex CR structure.