Existence of Cartesian vector fields for constrained Lagrangian systems

Let $\textsc{Q}$ be the configuration space, let $\lambda_k$ be the functions defining the system, and let

$$\Lambda_j=\sum_{k=1}^N a_{jk}\lambda_k,\qquad a_{jk}=-a_{kj},$$

for $j=M+1,\ldots,N$. A vector field is called Cartesian if it generates the first-order system $\dot{\textbf{x}}=\textbf{v}(x)$ under the conditions $\Lambda_j(x)=0$, and it is Cartesian equivalent if a nonzero function $\kappa$ exists such that $\kappa\breve{\textbf{v}}$ is Cartesian. Cartesian-vector-field existence conjecture. There are solutions of the equations $\Lambda_j=0$ for $j=M+1,\ldots,N$ which generate a Cartesian or Cartesian-equivalent vector field that completely describes the behavior of the constrained Lagrangian system. This asserts the existence of solutions to the first-order partial differential equations defining the constraints, but the source does not specify conditions guaranteeing such solutions or establish their general existence.