## Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I: General theory and $$\tau$$-function.(English)Zbl 1194.34167

Summary: A general theory of monodromy preserving deformation is developed for a system of linear ordinary differential equations $$dY/dx=A(x)Y$$, where $$A(x)$$ is a rational matrix. The non-linear deformation equations are derived and their complete integrability is proved. An explicit formula is found for a 1-form $$\omega$$, expressed rationally in terms of the coefficients of $$A(x)$$, that has the property $$d\omega =0$$ for each solution of the deformation equations. Examples corresponding to the “soliton” and “rational” solutions are discussed.

### MSC:

 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 34A30 Linear ordinary differential equations and systems
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### References:

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