On the Class of Potentials with Trivial Monodromy

Let Omega be a simple-connected domain, Z = {z(k) is an element of Omega, k = 1, N} (N <= infinity), and let TM(Omega, Z) be a set of functions meromorphic in Omega and satisfying at each its pole z(k) the trivial monodromy condition. The criterion for trivial monodromy is well known (it was established by Duistermaat and Gru<spacing diaeresis> nbaum in 1987). However, this criterion is of a local nature. It is impossible to extract from it any information about the structure of the set TM(Omega, Z). In this paper, we obtaine an explicit description of the set TM(Omega, Z) for N < infinity. In the case N = infinity, we establishe a certain analogue of theMittag-Leffler theorem: for any sequence of natural numbers {m(k)} there exists a function q is an element of TM(Omega, Z), which at each point z(k) satisfies the Duistermaat-Grunbaum condition with indicator mk.