Lyapunov equation on 2-jet [Byzantine]

Reminding myself that I jotted down the following sequence of steps back in May:

1. Write the partial differential equations for a Lyapunov function v as an exterior differential system in the 1-jet of v, using the contact form.
2. Write the equation in the 2-jet of v.
3. Then write it again in the 2-jet, using the connection forms, to bring in the Riemannian geometry of the manifold.
4. Now write the partial differential equation of Kalman filtering in the 2-jet of the total space.
5. Repeat using the connection forms.
   

In the last month or two I have made very little progress beyond step 2. This is not encouraging. Progress on solving a partial differential equation that defines a common Lyapunov function for two vector fields a and b was also limited.