Nahm equations
In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the "Nahm transform" – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators. Deep study of the Nahm equations was carried out by Nigel Hitchin and Simon Donaldson. Conceptually, the equations arise in the process of infinite-dimensional hyperkähler reduction. Among their many applications we can mention: Hitchin's construction  of monopoles, where this approach is critical for establishing nonsingularity of monopole solutions; Donaldson's description of the moduli space of monopoles; and the existence of hyperkähler structure on coadjoint orbits of complex semisimple Lie groups, proved by Peter Kronheimer, Olivier Biquard, and A.G. Kovalev. Equations. Let formula_1 be three matrix-valued meromorphic functions of a complex variable formula_2. The Nahm equations are a system of matrix differential equations together with certain analyticity properties, reality conditions, and boundary conditions. The three equations can be written concisely using the Levi-Civita symbol, in the form More generally, instead of considering formula_5  by formula_5 matrices, one can consider Nahm's equations with values in a Lie algebra formula_7. Additional conditions. The variable formula_2 is restricted to the open interval formula_9, and the following conditions are imposed: Nahm–Hitchin description of monopoles. There is a natural equivalence between Lax representation. The Nahm equations can be written in the Lax form as follows. Set then the system of Nahm equations is equivalent to the Lax equation As an immediate corollary, we obtain that the spectrum of the matrix formula_26 does not depend on formula_2. Therefore, the characteristic equation which determines the so-called spectral  curve in the twistor space formula_29is invariant under the flow in formula_2.