  We study singular integral operators induced by $3$-dimensional
Calder\'on-Zygmund kernels in the Heisenberg group. We show that if such an
operator is $L^{2}$ bounded on vertical planes, with uniform constants, then it
is also $L^{2}$ bounded on all intrinsic graphs of compactly supported
$C^{1,\alpha}$ functions over vertical planes. In particular, the result
applies to the operator $\mathcal{R}$ induced by the kernel $$\mathcal{K}(z) =
\nabla_{\mathbb{H}} \| z \|^{-2}, \quad z \in \mathbb{H} \setminus \{0\},$$ the
horizontal gradient of the fundamental solution of the sub-Laplacian. The
$L^{2}$ boundedness of $\mathcal{R}$ is connected with the question of
removability for Lipschitz harmonic functions. As a corollary of our result, we
infer that the intrinsic graphs mentioned above are non-removable. Apart from
subsets of vertical planes, these are the first known examples of non-removable
sets with positive and locally finite $3$-dimensional measure.
