Overdetermined 2D Systems Invariant in One Direction and Their Transfer   Functions
  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. 
Andrey Melnikov, Victor Vinnikov