= Solution
Using the <left-invariant frame of the real Heisenberg group>, horizontality gives
$$
\dot x=\alpha,\qquad \dot y=\beta,\qquad \dot z=x\beta.
$$
For an arbitrary starting point $(x_0,y_0,z_0)$,
$$
\boxed{x(t)=x_0+\int_0^t\alpha(s)\,ds,\qquad
y(t)=y_0+\int_0^t\beta(s)\,ds,\qquad
z(t)=z_0+\int_0^t\left(x_0+\int_0^s\alpha(r)\,dr\right)\beta(s)\,ds.}
$$
This is the control equation for the <Heisenberg horizontal distribution>. The final coordinate records a signed area-like integral, so changing the path in the $(x,y)$ plane can change the endpoint in the central direction.
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