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Related question (The system is not limited to integrable model, so I think this question is different)

As far as I know in quantum mechanics, exact solutions for multivariable systems (from partial differential equation, than ordinary differential equation), e.g., the hydrogen atom and Hooke atom, are obtained from separation of variable method. Is there any multivariable example in physics (doesn't have to be quantum), the exact solution can be obtained without using separable of variable method?

ElementSegment
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There is the method of characteristics, for example: d'Alambert formula and other method like Poisson kernel , Kirchhoff formula,...

ps: see, course of higher mathematics volume II and IV, by V.Smirnov. Lectures on Partial Differential equations, by I.G.Petrovsky.

Ordinary differential equation, by E.L.Ince.

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The Tiler
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