The Neumann-Rosochatius integrable system and multi-spin single spike string solutions on R x S5. Part I

We study rigid string solutions rotating in AdS5xS5 background. For particular values of the parameters of the solutions we find multi-spin solutions corresponding to giant magnons and single spike strings. We present an analysis of the dispersion relations in the case of three-spin solutions distributed only in S5. The possible relation of these string solutions to gauge theory operators and spin chains are briefly discussed. We find the solutions to the resulting Neumann-Rosochatius integrable system. (authors)