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It is well known that the Painlev\'e equations can formally degenerate toautonomous differential equations with elliptic function solutions in suitablescaling limits. A way to make this degeneration rigorous is to apply Deift-Zhousteepest-descent techniques to a Riemann-Hilbert representation of a family ofsolutions. This method leads to an explicit approximation formula in terms oftheta functions and related algebro-geometric ingredients that is difficult todirectly link to the expected limiting differential equation. However, theapproximation arises from an outer parametrix that satisfies relatively simpleconditions. By applying a method that we learned from Alexander Its, it ispossible to use these simple conditions to directly obtain the limitingdifferential equation, bypassing the details of the algebro-geometric solutionof the outer parametrix problem. In this paper, we illustrate the use of thismethod to relate an approximation of the algebraic solutions of thePainlev\'e-III (D$_7$) equation valid in the part of the complex plane wherethe poles and zeros of the solutions asymptotically reside to a form of theWeierstrass equation.

symmetry, integrability and geometry: methods and applications

Differential Equations for Approximate Solutions of Painlev\'e  Equations: Application to the Algebraic Solutions of the Painlev\'e-III  $({\rm D}_7)$ Equation