New non-integrability conditions are obtained for natural systems with exponential interaction (generalized Toda lattices) which allows us to close the classification problem of integrable systems of this kind. A notion of integrability is considered that means the presence of a complete set of first integrals being complex-meromorphic functions in the phase variables and, thus, more general than the integrability in the sense of Birkhoff (i.e. the presence of a complete set of first integrals being polynomials in momenta and exponents of coordinates with some coefficients) discussed earlier. However, it is established that the generalized Toda lattices integrable in our sense are also integable in Birkhoff sense.
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