Niederlinski's theorem essentially provides a method for recognizing the lack of structural stabi-lizability in multiloop integral control systems. The purpose of this study is to complement Niederlinski's original theorem by relaxing some of the restrictions it encounters for its application. The fact that Niederlinski's index can be applied to the plants exhibiting time delays and unstable poles in their diagonal (controlled) elements is noted in this paper. It is also shown that Niederlinski's original assumption, that in order to apply the NI to the problem of integral instability all the elements of the loop transfer matrix must be strictly proper, can in fact be relaxed to the condition that all the triangular elements of the loop transfer matrix must be strictly proper. Thus even in a case where some elements of plant transfer matrix, and consequently of loop transfer matrix are not strictly proper, the NI still remains applicable if in fact the triangular elements of the loop transfer matrix are strictly proper. This same simplification of conditions is likewise shown to be applicable in the case of Relative Gain Array (RGA). Finally, an application of pairing criteria, by use of NI and RGA, to unstable plants under multiloop integral control is proposed.