Definition: A pseudo-group is a collection $\mathcal{G}$ of (locally defined) invertible smooth diffeomorphisms of a manifold $M$. The simplest example of a pseudo-group is the collection of all local diffeomorphism of a manifold $M$.
Example: Consider the infinite-dimensional Lie pseudo-group
$$X=x,\qquad Y=y\,f(x)+\phi (x),\qquad Z=z(f(x))^x+\psi(x),$$
where $f,\phi,\psi\in C^\infty (\mathbb{R})$ and $f(x)>0$.
Obviously, $X,Y$ and $Z$ are the target coordinates. Indeed, the pseudo-group acts on the coordinate $(x,y,z)$ and transforms it to the target coordinate $(X,Y,Z)$.
My questions:
what are the group parameters??
If we split the cotangent bundle into horizontal and vertical forms, what are the vertical forms?
Reference
Olver, Peter J.; Pohjanpelto, Juha; Valiquette, Francis, On the structure of Lie pseudo-groups, SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 077, 14 p. (2009). ZBL1241.58008.