Application of Darboux's theorem to magnetic field line flows

In this famous paper by Cary and Littlejohn on noncanonical Hamiltonian mechanics and its application to magnetic field line flow, they claim that as a result of Darboux's theorem, it is always possible to cast the vector potential one form $$\mathbf{A}\cdot d\mathbf{x} = A_idx^i$$ in canonical form $$p\,dq - \mathcal{H}(p,q,t)\,dt$$. I can see that this is possible by making a gauge choice that eliminates one of the terms from the vector potential one form, but how is this a corollary of Darboux's theorem?

Given a $$1$$-form $$A~=~\sum_{I=1}^n A_I \mathrm{d}x^I \tag{1}$$ of constant rank in an odd-dimensional manifold $$M$$, it follows from the Darboux theorem that there exists a local coordinate system $$(\underbrace{q^1, \ldots, q^r}_{\text{positions}} ;\underbrace{p_1,\ldots, p_r}_{\text{momenta}}; \underbrace{c^1, \ldots, c^{n-1-2r}}_{\text{Casimirs}};\underbrace{t}_{\text{time}})\tag{2}$$ in a neighborhood $$U\subseteq M$$ such that the $$1$$-form $$A$$ is of the form $$A|_U ~=~ \sum_{i=1}^r p_i \mathrm{d}q^i -H \mathrm{d}t,\tag{3}$$ where $$H$$ is a constant (either $$0$$ or $$-1$$).