Distributional solutions of Burgers type equations for intrinsic graphs in Carnot groups of step 2

We prove that in arbitrary Carnot groups $mathbb G$ of step 2, with a splitting $mathbb G=mathbb Wcdotmathbb L$ with $mathbb L$ one-dimensional, the graph of a continuous function $varphicolon Usubseteq mathbb Wto mathbb L$ 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 Holder regularity along vertical directions.