In this paper, we continue the investigation of the constant astigmatism equation zyy + (1/z)xx + 2 = 0. We newly interpret its solutions as describing spherical orthogonal equiareal patterns, which links them to principal stress lines under the Tresca yield condition on the sphere. By extending the classical Bianchi superposition principle for the sine-Gordon equation, we show how to generate an arbitrary number of solutions by algebraic manipulations. Finally, we show that slip line fields on the sphere are determined by the sine-Gordon equation.

We introduce an algebraic formula producing infinitely many exact solutions of the constant astigmatism equation z yy +(1/z) xx +2 = 0 from a given seed. A construction of corresponding surfaces of constant astigmatism is then a matter of routine. As a special case, we consider multisoliton solutions of the constant astigmatism equation defined as counterparts of famous multisoliton solutions of the sine-Gordon equation. A few particular examples are surveyed as well.

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Abstract. We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation z yy + (1/z) xx + 2 = 0. The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter λ = ±1. It is also a nonlocal symmetry.

Abstract. For the constant astigmatism equation, we construct a system of nonlocal conservation laws (an abelian covering) closed under the reciprocal transformations. We give functionally independent potentials modulo a Wronskian type relation.