eq1 = sigrrR == ((2 C2 + (C4 minusnu r^minusnu + C3 plusnu r^plusnu)/
      r^2) (1 - 2 G alpha -
      2 E^(t gamma) G t alpha ExpIntegralEi[-t gamma]))/(2 G t);
eq2 = sigttR == ((2 C2 + (
      C4 (-1 + minusnu) minusnu r^minusnu +
       C3 (-1 + plusnu) plusnu r^plusnu)/r^2) (1 - 2 G alpha -
      2 E^(t gamma) G t alpha ExpIntegralEi[-t gamma]))/(2 G t);
eq3 = sigzzR == -(1/(
     24 G^3 K r^2 t z)) (2 G -
      3 K) (3 K n r^2 (h (rhors - g rhowater) (3000 + z) -
         z (-2 p + rhors (3000 + z))) +
      G (-2 h n r^2 (rhors - g rhowater) (3000 + z) +
         2 z (12 C2 K r^2 +
            3 K (C4 minusnu^2 r^minusnu + C3 plusnu^2 r^plusnu) +
            n r^2 (-2 p + rhors (3000 + z))))) (-1 + 2 G alpha +
      2 E^(t gamma) G t alpha ExpIntegralEi[-t gamma]);
system = {eq1, eq2, eq3};
Solve[system, {C2, C3, C4}]