In the seminar we discuss isospectral symmetries of matrix Schrödinger operators with periodic potentials (with a possible generalization to the quasi-periodic case, and to the stochastic case). Spectral theory of the (matrix) Schrödinger operator is an important mathematical tool in such areas as functional analysis, dynamical systems, integrable models etc. Following A.P. Veselov and A.B. Shabat “…the KdV theory is exactly the theory of isospectral symmetries of type (1) of the Schrodinger operator and, therefore, could have arisen independently within the framework of spectral theory”. We will study the dressing chain of associated matrix Riccati equations. Our main finding in this case is a special (local) symmetry between diagonal operators and operators with a non-trivial interaction. We also consider possible applications of these results in the theory of integrable systems, analysis of stability of mechanical systems, design of periodic structures with desired spin/polarization transport (1-D anisotropic photonic crystals).