Integrable systems on symmetric spaces from a quadratic pencil of Lax operators

The article surveys the recent results on integrable systems arising from quadratic pencil of Lax operator L, with values in a Hermitian symmetric space. The counterpart operator M in the Lax pair defines positive, negative and rational flows. The results are illustrated with examples from the A.III symmetric space. The modeling aspect of the arising higher order nonlinear Schr\"odinger equations is briefly discussed.Comment: 8 pages, Dedicated to Professor Vladimir Gerdjikov on the occasion of his 75th birthday. arXiv admin note: text overlap with arXiv:2301.0722