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Thank you very much for the effort. Using Pari/GP I have the full set of (even) solutions up to $x \approx 100$ Interesting, I thought there are only even solutions... But well, my concern was - - 1) to get some analytical answer on...
Just for the record: the solution is the geometric series in x, applicable for x from - infinity to 1 (exclusively); it's just a matrix-summation-method.       So I do know now what was happening - many thanks for your answers/help !