I have noticed now that your list of unknowns has a duplicate oma3:

{pia1, pib1, oma1, omb1, pia3, oma3, pib3, oma3}

Your system has 8 equations in 7 unknowns.

Ciao Gianluca, grazie dalla tua osservazione. Purtroppo, malgrado avessi cancellato la incognita doppia, non si è risolto il problema. Vedró di ridurre il numero sia di equazioni che d'incognite.

Dopo tutto Mathematica è uno strumento molto potente per la manipulazione simbolica. Ne trarrò profitto.

Buona sera,

Gustavo.

Fixing the RN error, and replacing N with n because N[] is a built-in function, Solve alone still produces no general solution, but adding MaxExtraConditions->All, after a pretty long time (minutes), produces a whole list. Which suggests you may get your answer if you can add some assumptions to your input.

Solve[{
  n^2 pia1 R == pib1/2,
  1/48 n - 3 R n^4 pia1 + R n^4 pia3 == -3 pib1 + pib3,
  (R n^6 pia1)/384 - (R n^6 pia3)/768 == pib1/384 - pib3/768,
  (R n^10 pia1)/1474560 - (R n^10 pia3)/2211840 == 
   pib1/1474560 - pib3/2211840,
  0 == newalb10 (pib1 + omb1 betr) + 
    2 epsb (pib3 ch3r/chr + omb3 sh3r/chr),
  0 == newala10 (pia1 - oma1 bett) + 
    2 epsa (pia3 ch3t/cht + omb3 sh3t/cht),
  (R n^10 pia1)/1474560 - (R n^10 pia3)/2211840 == 
   pib1/1474560 - pib3/2211840,
  -((R n^12 pia1)/176947200) + (R n^12 pia3)/
    247726080 == -(pib1/176947200) + pib3/247726080},
 {pia1, pib1, oma1, omb1, pia3, oma3, pib3}, 
 MaxExtraConditions -> All]

Thank you very much GianLuca. Yes, I will check that. On a first glimpse, it seems that omb3 is lacking. Hope this is all the problem

It is a cumbersome system of equations and it is very easy not to count the number of the equations correctly.

Take care, I am very grateful,

Gustavo