Algorithm for determining ${\mathcal{C}}^{\mathrm{\infty }}$-structures

This map of content outlines the systematic method for constructing a cinf-structure for a given ODE, transforming the search for vector fields into a search for scalar functions.

Concepts

• canonical cinf-structure: The simplified, triangular representative form of the structure used for the search.
• recursive prolongation formula: The formula used to generate the vector field components from the $\lambda$ functions.
• lambda-generated vector fields: The vector fields explicitly constructed from the $\lambda$ functions.
• determining equations for cinf-structures: The system of PDEs derived from the involution conditions that the $\lambda$ functions must satisfy.
• ansatz for cinf-structures: The strategy of assuming simplified functional forms for ${\lambda }_i$ to solve the determining equations.