The Euler Equations of Spatial Gasdynamics and the Integrable Heisenberg Spin Equation By W. K. Schief and C. Rogers The Euler equations for an inviscid and thermally nonconducting gas are investigated with a view to isolating particular integrable structure. A natural physical constraint is imposed and it is demonstrated that, in the generic case, the steady Euler equations are equivalent to an integrable Heisenberg spin equation subject to a volume-preserving constraint. A limiting case in which this constraint is not present is indicated. 1. Introduction A long-standing problem in hydrodynamics which generalizes a well-known problem posed by Hamel in [1] is concerned with the determination of all steady hydrodynamic flows which are uniquely determined by their streamline patterns. This generalization was originally formulated by Gilbarg [2] and subsequently investigated by Prim [3] who established that any steady spatial hydrodynamic flow is unique (up to scaling) unless its speed is constant along individual streamlines. Thus, the resolution of Gilbarg’s problem amounts to the determination of all so-called “constant speed” flows. The latter were subsequently investigated by Howard [4], Wasserman [5], and Marris [6]. In [7, 8], it was established that, remarkably, constant speed flows are governed by the integrable Heisenberg spin equation [9] subject to a geometric constraint. The significance of the mathematically equivalent analog of constant speed Address for correspondence: W. K. Schief, School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia; e-mail: schief@maths.unsw.edu.au DOI: 10.1111/j.1467-9590.2011.00539.x 407 STUDIES IN APPLIED MATHEMATICS 128:407–419 C  2011 by the Massachusetts Institute of Technology