This time

ClearAll;
H = {{0, 0, OmP}, {0, 2*d, OmC}, {OmP, OmC, 2*d}};
r = {{r11, r12, r13}, {r21, r22, r23}, {r31, r32, r33}};
Kommutator = (i/2)*(H.r - r.H);
G = {{G1*r33, 0, -(1/2)*r13*(G1 + G2)}, {0, G2*r33, -(1/2)*r23*(G1 + G2)}, {-(1/2)*r31*(G1 + G2), 
        -(1/2)*r32*(G1 + G2), -r33*(G1 + G2)}};
Nul = {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}};



In[81]:= Solve[{Kommutator + G == Nul}, {r11, r12, r13, r21, r22, r23, r31, r32, r33}]

During evaluation of In[81]:= Solve::svars: Equations may not give solutions for all "solve" variables. >>

Out[81]= {{r11 -> -(((-4 d^2 G1^3 - 8 d^2 G1^2 G2 - 4 d^2 G1 G2^2 + 
       16 d^4 G1 i^2 - 8 d^2 G1 i^2 OmC^2 + G1 i^2 OmC^4 + 
       4 d^2 G1 i^2 OmP^2 + G1 i^2 OmC^2 OmP^2 + G2 i^2 OmC^2 OmP^2 + 
       G2 i^2 OmP^4) r13)/(
    2 d i OmP (-2 d G1^2 - 2 d G1 G2 + 4 d^2 G1 i - G1 i OmC^2 - 
       G2 i OmP^2))), 
  r12 -> -(((-2 d G1^2 OmC^2 - 2 d G1 G2 OmC^2 + 4 d^2 G1 i OmC^2 - 
       G1 i OmC^4 - 2 d G1 G2 OmP^2 - 2 d G2^2 OmP^2 - 
       G1 i OmC^2 OmP^2 - G2 i OmC^2 OmP^2 - G2 i OmP^4) r13)/(
    2 d OmC (-2 d G1^2 - 2 d G1 G2 + 4 d^2 G1 i - G1 i OmC^2 - 
       G2 i OmP^2))), 
  r21 -> -(((2 d G1^2 OmC^2 + 2 d G1 G2 OmC^2 + 4 d^2 G1 i OmC^2 - 
       G1 i OmC^4 + 2 d G1 G2 OmP^2 + 2 d G2^2 OmP^2 - 
       G1 i OmC^2 OmP^2 - G2 i OmC^2 OmP^2 - G2 i OmP^4) r13)/(
    2 d OmC (-2 d G1^2 - 2 d G1 G2 + 4 d^2 G1 i - G1 i OmC^2 - 
       G2 i OmP^2))), 
  r22 -> (OmP (4 d^2 G1^2 G2 + 8 d^2 G1 G2^2 + 4 d^2 G2^3 - 
      4 d^2 G2 i^2 OmC^2 - G1 i^2 OmC^4 - G1 i^2 OmC^2 OmP^2 - 
      G2 i^2 OmC^2 OmP^2 - G2 i^2 OmP^4) r13)/(
   2 d i OmC^2 (-2 d G1^2 - 2 d G1 G2 + 4 d^2 G1 i - G1 i OmC^2 - 
      G2 i OmP^2)), 
  r23 -> (OmP (2 d G1 G2 + 2 d G2^2 - G1 i OmC^2 - G2 i OmP^2) r13)/(
   OmC (2 d G1^2 + 2 d G1 G2 - 4 d^2 G1 i + G1 i OmC^2 + G2 i OmP^2)),
   r31 -> -(((-2 d G1^2 - 2 d G1 G2 - 4 d^2 G1 i + G1 i OmC^2 + 
       G2 i OmP^2) r13)/(-2 d G1^2 - 2 d G1 G2 + 4 d^2 G1 i - 
     G1 i OmC^2 - G2 i OmP^2)), 
  r32 -> -((
    OmP (2 d G1 G2 + 2 d G2^2 + G1 i OmC^2 + G2 i OmP^2) r13)/(
    OmC (2 d G1^2 + 2 d G1 G2 - 4 d^2 G1 i + G1 i OmC^2 + 
       G2 i OmP^2))), 
  r33 -> (2 (d G1 + d G2) i OmP r13)/(
   2 d G1^2 + 2 d G1 G2 - 4 d^2 G1 i + G1 i OmC^2 + G2 i OmP^2)}}

Mathematica (10.4.1.0) does not solve for r13 but r13 is a linear factor to all other (r11, r12, r21, r22, r23, r31, r32, r33) terms, so it appears that r13 has the role of a norm, hasn`t it? And if so, it is not Mathematica's fault, but a hint to you to normalize your expressions, please.

Just to mention it, more readable expressions for Nul are

In[83]:= DiagonalMatrix[{0, 0, 0}]
Out[83]= {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

In[85]:= ConstantArray[0, {3, 3}]
Out[85]= {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

In[86]:= DiagonalMatrix[ConstantArray[0, {3}]]
Out[86]= {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

In[91]:= 0 IdentityMatrix[3]
Out[91]= {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

Then click the error message (ref/message/Solve/svars) to read

This message is generated when the number of variables listed in the second argument of Solve is greater than the number of variables that could be determined from the equations.

That's clear enough. Next thing you could do, leave this or that variable out to see, whether is makes an effect:

Solve[{Kommutator + G == Nul}, {r11, r12, r13, r21, r22, r23, r31, r32 (*,r33 *)}]

Now r33 is a linear factor and no warning appears.

Solve[{Kommutator + G == Nul}, {r11, r12,(* r13, *) r21, r22, r23, r31, r32, r33 }]

Now r13 is a linear factor. What does it mean?

My mathematica code is not the problem, right?

You need to add a norm condition to the equation to get rid of the warning message.

Solve[{Kommutator + G == Nul}, {(* r11, *)r12, r13, r21, r22, r23, r31, r32, r33 }]

All are solved with the skipped variable as linear factor: With other words, the missing condition fixes the up-to-now free linear factor.