  Let F be a smooth real manifold with a linear connection in the tangent
bundle. How can we extend the coefficients of the connection to bi-differential
operators that incorporate the original structure at zero order? Take a
constant mapping of F to a point. Suppose that the point belongs to another
manifold M^n. Consider a fibre bundle over M^n with fibres F. Denote the
projection of the total space to M^n by \pi. Construct the space of infinite
jets $J^\infty(\pi)$ of sections for this bundle. The
$C^\infty(J^\infty(\pi))$-module of \pi-vertical evolutionary fields on the jet
space is the analogue of the $C^\infty(F)$-module of vector fields on F, which
now `feels' the presence of the base M^n. We define and classify the matrix
linear differential operators whose images in the module of evolutionary fields
are closed w.r.t. commutation. We define the compatibility for such operators
with collective commutation closure. We prove that, under reparameterizations
in their domain, the bi-differential structural constants, which describe the
commutation of the fields in the sum of their images, obey an analogue of the
transformation rule for Christoffel symbols. We show that the new structure is
the connection in the pair of Lie algebras (the domain and the sum of images of
the operators) related by homomorphisms (points in the linear space spanned by
the operators). The application of this theory to geometry of PDE solves a
long-standing problem. We give a complete description of symmetry algebras for
2D Toda chains associated with semi-simple Lie algebras: we obtain all symmetry
generators and calculate all commutation relations. As a by-product, we expose
a geometric scheme for derivation of bi-Hamiltonian KdV-type hierarchies
related to root systems.
