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I believe I have found the answer. I think it works like this, but I still have to verify it. In case of any other people that might have the same question, I will outline it here. It relies on the following proposition (which can be found in, for example, the book I cited in my question):

Proposition. Let $\mathcal{C}$ be the Cartan connection on $J^\infty(\pi)$ and denote with $j^\infty(s)$ the infinite jet of some section $s$. A submanifold of $J^\infty(\pi)$ is a maximal integral manifold of $\mathcal{C}$ if and only if it is the graph of $j^\infty(s)$ for some section $s$.

Now take some evolutionary vector field $\partial_\varphi$. Then $\partial_\varphi$ is an infinitesimal automorphism of the Cartan distribution, i.e. $[\partial_\varphi, X] \in \mathcal{CX}(\pi)$ whenever $X \in \mathcal{CX}(\pi)$ (where $\mathcal{CX}(\pi)$ is the space of vector fields whose values lie in $\mathcal{C}$). This implies (I think) that its flow $F$ leaves $\mathcal{C}$ invariant, i.e. if $\theta \in J^\infty(\pi)$ then $(F_t)_*\mathcal{C}_\theta = \mathcal{C}_{F_t(\theta)}$. Therefore it maps maximal integral manifolds to maximal integral manifolds.

Take a section $s$. When restricted to the jet of $s$, Then the equation argument above becomes $(F_t)_*\mathcal{C}_{j_x^\infty(s)} = \mathcal{C}_{F_t({j_x^\infty(s)})}$. This implies (I believe) that the image of the map $x \mapsto F_t(j_x^\infty(s))$ is another maximal integral manifold of $\mathcal{C}$. Therefore, it comes from some other section $s_t$, i.e. there is a section $s_t$ such that $F_t(j_x^\infty(s)) = j^\infty_x(s_t)$ for all $x$. In this way, the evolutionary derivative determines a flow $s_t$ F$, when restricted to the jet of some section$s$becomes (the jet of) another section$s$. s_t$. Moreover, denote with $F_{t,\sigma}^i$ the $\sigma,i$-component of $F_t$, where $\sigma$ is a multi-index. Then the equation that determines that $F$ is a flow of $\partial_\varphi$ is $\left.\frac{d}{dt}\right|_{t=0}F^i_{t,\sigma}(\theta) = (\partial_\varphi)^i_\sigma(\theta) = (D_\sigma\varphi^i)(\theta)$ for $\theta \in J^\infty(\pi)$. When restricted to $j^\infty(s)$, this finally becomes

$$\left.\frac{d}{dt}\right|_{t=0}\frac{\partial^{|\sigma|}s_t^j}{\partial x^\sigma}(x) = (D_\sigma \varphi^i)(j^\infty(s)),$$

which is the equation that inspired my question.

2 deleted 1 characters in body

I believe I have found the answer. I think it works like this, but I still have to verify it. In case of any other people that might have the same question, I will outline it here. It relies on the following proposition (which can be found in, for example, the book I cited in my question):

Proposition. Let $\mathcal{C}$ be the Cartan connection on $J^\infty(\pi)$ and denote with $j^\infty(s)$ the infinite jet of some section $s$. A submanifold of $J^\infty(\pi)$ is a maximal integral manifold of $\mathcal{C}$ if and only if it is the graph of $j^\infty(s)$ for some section $s$.

Now take the generating section $\varphi$ of some evolutionary vector field $\partial_\varphi$. Then $\partial_\varphi$ is an infinitesimal automorphism of the Cartan distribution, i.e. $[\partial_\varphi, X] \in \mathcal{CX}(\pi)$ whenever $X \in \mathcal{CX}(\pi)$ (where $\mathcal{CX}(\pi)$ is the space of vector fields whose values lie in $\mathcal{C}$). This implies (I think) that its flow $F$ leaves $\mathcal{C}$ invariant, i.e. if $\theta \in J^\infty(\pi)$ then $(F_t)_*\mathcal{C}_\theta = \mathcal{C}_{F_t(\theta)}$.

Take a section $s$. When restricted to the jet of $s$, the equation above becomes $(F_t)_*\mathcal{C}_{j_x^\infty(s)} = \mathcal{C}_{F_t({j_x^\infty(s)})}$. This implies (I believe) that the image of the map $x \mapsto F_t(j_x^\infty(s))$ is another maximal integral manifold of $\mathcal{C}$. Therefore, it comes from some other section $s_t$, i.e. $F_t(j_x^\infty(s)) = j^\infty_x(s_t)$ for all $x$. In this way the evolutionary derivative determines a flow $s_t$ of the section $s$. Moreover, denote with $F_{t,\sigma}^i$ the $\sigma,i$-component of $F_t$, where $\sigma$ is a multi-index. Then the equation that determines that $F$ is a flow of $\partial_\varphi$ is $\left.\frac{d}{dt}\right|_{t=0}F^i_{t,\sigma}(\theta) = (\partial_\varphi)^i_\sigma(\theta) = (D_\sigma\varphi^i)(\theta)$ for $\theta \in J^\infty(\pi)$. When restricted to $j^\infty(s)$, this finally becomes

$$\left.\frac{d}{dt}\right|_{t=0}\frac{\partial^{|\sigma|}s_t^j}{\partial x^\sigma}(x) = (D_\sigma \varphi^i)(j^\infty(s)),$$

which is the equation that inspired my question.

1

I believe I have found the answer. I think it works like this, but I still have to verify it. In case of any other people that might have the same question, I will outline it here. It relies on the following proposition (which can be found in, for example, the book I cited in my question):

Proposition. Let $\mathcal{C}$ be the Cartan connection on $J^\infty(\pi)$ and denote with $j^\infty(s)$ the infinite jet of some section $s$. A submanifold of $J^\infty(\pi)$ is a maximal integral manifold of $\mathcal{C}$ if and only if it is the graph of $j^\infty(s)$ for some section $s$.

Now take the generating section $\varphi$ of some evolutionary vector field $\partial_\varphi$. Then $\partial_\varphi$ is an infinitesimal automorphism of the Cartan distribution, i.e. $[\partial_\varphi, X] \in \mathcal{CX}(\pi)$ (where $\mathcal{CX}(\pi)$ is the space of vector fields whose values lie in $\mathcal{C}$). This implies (I think) that its flow $F$ leaves $\mathcal{C}$ invariant, i.e. if $\theta \in J^\infty(\pi)$ then $(F_t)_*\mathcal{C}_\theta = \mathcal{C}_{F_t(\theta)}$.

Take a section $s$. When restricted to the jet of $s$, the equation above becomes $(F_t)_*\mathcal{C}_{j_x^\infty(s)} = \mathcal{C}_{F_t({j_x^\infty(s)})}$. This implies (I believe) that the image of the map $x \mapsto F_t(j_x^\infty(s))$ is another maximal integral manifold of $\mathcal{C}$. Therefore, it comes from some other section $s_t$, i.e. $F_t(j_x^\infty(s)) = j^\infty_x(s_t)$ for all $x$. In this way the evolutionary derivative determines a flow $s_t$ of the section $s$. Moreover, denote with $F_{t,\sigma}^i$ the $\sigma,i$-component of $F_t$, where $\sigma$ is a multi-index. Then the equation that determines that $F$ is a flow of $\partial_\varphi$ is $\left.\frac{d}{dt}\right|_{t=0}F^i_{t,\sigma}(\theta) = (\partial_\varphi)^i_\sigma(\theta) = (D_\sigma\varphi^i)(\theta)$ for $\theta \in J^\infty(\pi)$. When restricted to $j^\infty(s)$, this finally becomes

$$\left.\frac{d}{dt}\right|_{t=0}\frac{\partial^{|\sigma|}s_t^j}{\partial x^\sigma}(x) = (D_\sigma \varphi^i)(j^\infty(s)),$$

which is the equation that inspired my question.