# Understanding Noether's second theorem

Wikipedia writes that

"if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by $$k$$ arbitrary functions and their derivatives up to order $$m$$, then the functional derivatives of $$L$$ satisfy a system of $$k$$ differential equations."

Such a transformation may look like $$\delta\varphi = \text{i} \omega_a(x)T^a F[\varphi] + \text{i}\sum\limits_{j=1}^{m} F^{\mu_1..\mu_j}[\varphi](\partial_{\mu_1}..\partial_{\mu_j})\omega_a(x)T^a \quad a=1,..,k$$

They specify the parameters $$k$$ and $$m$$, however only state how $$k$$ is interpreted. What about $$m$$? How does the choice of $$m$$ affect Noether's second theorem?

Noether’s second theorem gives us strong identities, which constrain the form of the equations of motion and the current $$j^\mu$$.
In particular, for the case $$m=1$$, they show that
$$E^k F^a_k[\varphi] - \partial_\mu\big( E^k F^{a\mu}_k[\varphi] \big) \stackrel{!}= 0 \tag{1}$$
where $$E^k$$ are the equations of motion for the field $$\varphi_k$$ and the functionals $$F$$ as specified in the question. Different choices of $$m$$ will alter the explicit shape of (1). However,