On Nirenberg's non-embeddable CR structure

We study $mathfrak{X}$-valued CR functions of Theodoresco class $B^1$ on the Heisenberg group $mathbb{H}_1 = mathbb {R} times mathbb{C}$ equipped with the non embeddable CR structure discovered by L. Nirenberg [On a question of Hans Lewy. Russian Math. Surveys, 1974;29, 251-262] where $mathfrak{X}$ is an arbitrary complex Fréchet space. If $overline{L}_varphi equiv overline{L} + varphi partial /partial x$ is Nirenberg's perturbation of Lewy's operator $overline{L} = partial /partial overline{w} - i w partial /partial x$, we show that for every open neighborhood $U subset mathbb{H}_1$ of the origin, no two solutions $f^a : U to mathfrak{X}$, $a in { 1, 2 }$, of Theodoresco class $B^1$, to the tangential Cauchy-Riemann equations $overline{L}_varphi f = 0$ are functionally independent