$\tilde{\lambda }$-generated vector fields

Lambda-generated vector fields are the set of vector fields $\{X_1,\dots ,X_{m-1}\}$ explicitly constructed from an arbitrary set of scalar functions $\{{\lambda }_1,\dots ,{\lambda }_{m-1}\}$ using the recursive prolongation formula.
Observe that these fields do not necessarily form a ${\mathcal{C}}^{\mathrm{\infty }}$-structure. They only constitute a valid structure (forming an involutive distribution) if the generating functions ${\lambda }_i$ satisfy a specific system of PDEs known as the determining equations for cinf-structures.

Role in the Algorithm

These vector fields serve as the candidate basis (or ansatz) for a canonical cinf-structure in the algorithm for determining cinf-structures. They transform the problem of finding vector fields into a problem of finding scalar functions.

Mathematical Definition

For a chosen set of functions $\tilde{\lambda }=\{{\lambda }_1,\dots ,{\lambda }_{m-1}\}$, the associated vector fields are defined as:

where the coefficients ${\eta }_i^k$ are computed recursively using the operator $A$ and the functions ${\lambda }_1,\dots ,{\lambda }_i$ according to the recursive prolongation formula.