\(\(a = Exp[I*2*π/3]
\)\)

eqn1 := U1 == U

eqn2 := U2 == (a^2)*U

eqn3 := U3 == a*U

eqn4 := I*Y*(U1 - U2) - I*Y*(U1 - U3) == k*U1

eqn5 := -I*Y*(U1 - U2) + G*(U2 - U3) == k*U2

eqn6 := G*(U3 - U2) - I*Y*(U3 - U1) == k*U3

Solve[{eqn1, eqn2, eqn3, eqn4, eqn5, eqn6}, {Y, k}, {U1, U2, U3}]FullSimplify[%]

Pz = r*(i1^2) + r*(i2^2) + r*(i3^2);
podm = u*(i1 + i2 + i3) - Pdod;
lagrange = Pz - λ*podm;
Solve[{D[lagrange, i1] == 0, D[lagrange, i2] == 0, 
          D[lagrange, i3] == 0, podm == 0}, {i1, i2, 
          i3, λ}][[1]]