  In this work we develop a theory of Vessels. This object arises in the study
of overdetermined 2D systems invariant in one of the variables, which are
usually called time invariant. To each overdetermined time invariant 2D systems
there is associated a vessel, which is a collection of system operators
satisfying certain relations and vise versa. Such an invariance forces the
theory of vessels to resemble a constant (classical) 1D case and as a result
many notions are naturally redefined and most theorems are reproved in this
setting. The notion of transfer function and its connection to the
overdetermined 2D time invariant system (and the corresponding vessel) is one
of the topics of this work. It is well known that multiplicative structure of a
transfer function of a 1D system is closely connected to the decomposition of
the state space into invariant subspaces of the state operator and we
generalize this result to a wider class of functions. This class (denoted by
$\boldsymbol {\mathcal I}$) arises as a class of transfer functions, which
intertwine solutions of ODEs with spectral parameters. At the end we present
solution of factorization problems for finite dimensional case.
