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Using a bigraded differential complex depending on the CR and pseudohermitianstructure, we give a characterization of three-dimensional stronglypseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclideanspace $\mathbb{R}^n$ in terms of an integral representation of Weierstrasstype. Restricting to the case of immersions in $\mathbb{R}^4$, we studyharmonicity conditions for such immersions and give a complete classificationof CR-pluriharmonic immersions.

Isometric and CR pluriharmonic immersions of three dimensional CR manifolds in Euclidean spaces