Canonical $C^{\infty}$ -Structure

Definition
In the context of integrating an $m$ -th order ODE, a canonical $C^{\infty}$ -structure is a unique, simplified representative within an equivalence class of $C^{\infty}$ -structures. It is defined as an ordered collection of vector fields $⟨ X_{1}, \dots, X_{m} ⟩$ that possess a strict triangular form with normalized coefficients: for an ODE with variables $(x, u, u_{1}, \dots, u_{m - 1})$ , the vector fields take the following "canonical" form:

\begin{aligned} X_{1} & = \partial_{u} + η_{1}^{1} \partial_{u_{1}} + η_{1}^{2} \partial_{u_{2}} + \dots + η_{1}^{m - 1} \partial_{u_{m - 1}} \\ X_{2} & = \partial_{u_{1}} + η_{2}^{2} \partial_{u_{2}} + \dots + η_{2}^{m - 1} \partial_{u_{m - 1}} \\ X_{3} & = \partial_{u_{2}} + \dots + η_{3}^{m - 1} \partial_{u_{m - 1}} \\ ⋮ \\ X_{m} & = \partial_{u_{m - 1}} \end{aligned}

That is, if we write

X_{i} = ξ_{i} \partial_{x} + η_{i}^{0} \partial_{u} + η_{i}^{1} \partial_{u_{1}} + \dots,

the coefficients satisfy $ξ_{i} = 0$ (no $\partial_{x}$ component), and for the derivative components $η_{j}^{i}$ , all terms below the diagonal are zero, and the diagonal term is normalized to 1 (i.e., $η_{i - 1}^{i} = 1$ ).

Key Properties

Minimal Determination: A canonical $C^{\infty}$ -structure is completely determined by exactly $m - 1$ smooth functions ( $λ_{1}, \dots, λ_{m - 1}$ ). This reduces the complexity of finding a structure compared to general methods.
Uniqueness: Every $C^{\infty}$ -structure for a given ODE belongs to an equivalence class that contains exactly one canonical representative (and several solvable structures).
Integrability: Finding a canonical structure allows to integrate the ODE by splitting the problem into a sequence of completely integrable Pfaffian equations.

Canonical C∞-Structure

Canonical $C^{\infty}$ -Structure