Аннотация:We consider a special system of integral equations of convolution type with a monotoneconvex nonlinearity naturally arising when searching for stationary or limit states in variousdynamic models of applied nature, for example, in models of the spread of epidemics, and provetheorems stating the existence or absence of a nontrivial bounded solution with limits at ±∞depending on the values of these limits and on the structure of the matrix kernel of the system.We also study the uniqueness of such a solution assuming that it exists. Specific examples ofsystems whose parameters satisfy the conditions stated in our theorems are given.