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From non-polynomial invariants to non-Liouvillian first integrals
We present a method of finding non-Liouvillian first integrals of rational two-dimensional differential systems. The method is based on the existence of two independent invariants that satisfy a linear second-order ordinary differential equation with respect to one of the variables. We call systems with this property R-integrable. These invariants are not necessarily polynomial; they can be expressed via special functions. We demonstrate that if there exists a rational transformation that maps some Riccati system to a given rational system, then the latter is always R-integrable. We prove that the famous Van der Pol systems and their generalizations are not R-integrable. We find novel integrable polynomial Liénard differential systems without invariant algebraic curves. The related first integrals are expressible via the Airy functions. We prove that the Kudashev system arising in the theory of the Korteweg–de Vries equation is R-integrable.