A possibly soft question for you guys and gals. Say a system of analytic PDEs has been completed to involution (in the sense that it's geometric symbol has a Pommaret basis, or has vanishing Spencer/Koszul (co)homology). Is it then always locally solvable?

Just to include why I'm asking: Lie-Pseudo-groups are determined by an involutive system and their Lie-algebroids by its linearization. If the answer to the question is positive, then since the linearized equation has the same symbol module (at the identity) it also is locally solvable. I have not been able to find this (possible) fact in the literature.

A possibly soft question for you guys and gals. Say a system of analytic PDEs has been completed to involution (in the sense that it's geometric symbol has a Pommaret basis, or has vanishing Spencer/Koszul (co)homology). Is its linearization locally solvable?

Just to include why I'm asking: Lie-Pseudo-groups are determined by an involutive system and their Lie-algebroids by its linearization. If the answer to the question is positive, then since the linearized equation has the same symbol module (at the identity) it also is locally solvable. I have not been able to find this (possible) fact in the literature.

edit: I changed the title of the question as it was pretty bad.

A possibly soft question for you guys and gals. Say a system of analytic PDEs has been completed to involution (in the sense that it's geometric symbol has a Pommaret basis, or has vanishing Spencer/Koszul (co)homology). Is it then always locally solvable?

Just to include why I'm asking: Lie-Pseudo-groups are determined by an involutive system and their Lie-algebroids by its linearization. If the answer to the question is positive, then since the linearized equation has the same symbol module (at the identity) it also is locally solvable.

A possibly soft question for you guys and gals. Say a system of analytic PDEs has been completed to involution (in the sense that it's geometric symbol has a Pommaret basis, or has vanishing Spencer/Koszul (co)homology). Is it then always locally solvable?

Just to include why I'm asking: Lie-Pseudo-groups are determined by an involutive system and their Lie-algebroids by its linearization. If the answer to the question is positive, then since the linearized equation has the same symbol module (at the identity) it also is locally solvable. I have not been able to find this (possible) fact in the literature.