Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel-Wu conjecture. The associated integrable nonlinear partial differential equationpossesses a zero curvature representation, a third-order symmetry, and a nonlocal transformation to the sine-Gordon equation φ ξη = sin φ. We leave open the problem of finding a Bäcklund autotransformation and a recursion operator that would produce a local hierarchy.

We consider the problem whether a nonparametric zero-curvature representation can be embedded into a one-parameter family within the same Lie algebra. After introducing a computable cohomological obstruction, a method using the recursion operator to incorporate the parameter is discussed.

We present a criterion of reducibility of a zero curvature representation to a solvable subalgebra, hence to a chain of conservation laws. Namely, we show that reducibility is equivalent to the existence of a section of the generalized Riccati covering. Results are applied to conversion between Guthrie's and Olver's form of recursion operators.

We present a new approach to construction of recursion operators for multidimensional integrable systems which have a Lax-type representation in terms of a pair of commuting vector fields. It is illustrated by the examples of the Manakov-Santini system which is a hyperbolic system in N dependent and N +4 independent variables, where N is an arbitrary natural number, the six-dimensional generalization of the first heavenly equation, the modified heavenly equation, and the dispersionless Hirota equation.

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.