There exist an open neighbourhood $M\subset \mathbb R^3$ of $0$, a point $p=0\in M$, and a nowhere-vanishing smooth complex vector field $\bar L\in \Gamma(M,\mathbb C TM)$ such that the smooth complex line subbundle $T^{0,1}M:=\operatorname{span}_{C^\infty(M;\mathbb C)}\{\bar L\}\subset \mathbb C TM$ satisfies $T^{0,1}M\cap \overline{T^{0,1}M}=\{0\}$ and $[\Gamma(T^{0,1}M),\Gamma(T^{0,1}M)]\subset \Gamma(T^{0,1}M)$. Moreover, for every neighbourhood $V\subset M$ of $p$ and every $N\in\mathbb N$, there is no smooth embedding $F:V\to \mathbb C^N$ whose component functions $F_1,\dots,F_N:V\to\mathbb C$ satisfy $\bar L F_j=0$ for every $j\in\{1,\dots,N\}$. Equivalently, the smooth formally integrable abstract CR structure $T^{0,1}M$ of hypersurface type and CR dimension $1$ is not locally CR-embeddable at $p$.