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The present paper derives systems of partial differential equations thatadmit a quadratic zero curvature representation for an arbitrary realsemisimple Lie algebra. It also determines the general form of Hamilton'sprinciples and Hamiltonians for these systems and analyzes the linear stabilityof their equilibrium solutions in the examples of $\mathfrak{so}(3)$ and$\mathfrak{sl}(2,\mathbb{R})$.

Integrable G-Strands on semisimple Lie groups