  We prove that in arbitrary Carnot groups $\mathbb G$ of step 2, with a
splitting $\mathbb G=\mathbb W\cdot\mathbb L$ with $\mathbb L$ one-dimensional,
the graph of a continuous function $\varphi\colon U\subseteq \mathbb W\to
\mathbb L$ is $C^1_{\mathrm{H}}$-regular precisely when $\varphi$ satisfies, in
the distributional sense, a Burgers' type system $D^{\varphi}\varphi=\omega$,
with a continuous $\omega$. We stress that this equivalence does not hold
already in the easiest step-3 Carnot group, namely the Engel group. As a tool
for the proof we show that a continuous distributional solution $\varphi$ to a
Burgers' type system $D^{\varphi}\varphi=\omega$, with $\omega$ continuous, is
actually a broad solution to $D^{\varphi}\varphi=\omega$. As a by-product of
independent interest we obtain that all the continuous distributional solutions
to $D^{\varphi}\varphi=\omega$, with $\omega$ continuous, enjoy $1/2$-little
H\"older regularity along vertical directions.
