Matrix equation with Mathematica

Question

I'm trying to solve the equation H1 = S.H2-H2.S where H1 and H2 are known 10x10 matrices depending on two parameters J1,J2 meanwhile S has to be determined.

In my try, I do

1. Define matrices

    H1 = J1/2*{{-1/2, 1, 0, 0, 0, 0, 0, 0, 0, 0},
    {1, -1/2, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, -1/2, 1, 0, 0, 0, 0, 0, 0},
    {0, 0, 1, -1/2, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, -1, 0, 1, 1, 0, 0},
    {0, 0, 0, 0, 0, -1, 1, 1, 0, 0},
    {0, 0, 0, 0, 1, 1, -1, 0, 0, 0},
    {0, 0, 0, 0, 1, 1, 0, -1, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 1, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}};
   

and

H2 = J2/2*{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, -1, 0, 0, 0, 1, 1},
{0, 0, 0, 0, 0, -1, 0, 0, 1, 1},
{0, 0, 0, 0, 0, 0, 1, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 1, 0, 0},
{0, 0, 0, 0, 1, 1, 0, 0, -1, 0},
{0, 0, 0, 0, 1, 1, 0, 0, 0, -1}};

2. Define S matrix as S = Array[X, {10, 10}];
3. Solve it: Solve[S.H1 - H1.S == H2, Flatten[S]]

The final result is an empty array {} which doesn't depend on J1,J2.

My question is: does it mean that the matrix S is an empty array because there is no solution or am I doing something wrong? Because I was expecting something depending on the parameters J1,J2.

I'd appreciate a lot comments on my issue!

Thanks :-)