The system of equations has multiple solutions that depend on the values of $H$, $F$, and $I$. The solutions given are quite complex and involve square roots and multiple possible outcomes for each variable.

Since you require the variables $A, B, C, D, E, F, G, H, I$ to be distinct integers between 1 and 9, we can impose these restrictions and search for the valid solutions.

Let me simplify this and calculate the integer solutions.The valid solution where the values are distinct integers between 1 and 9 is:

$A = 3, \quad B = 9, \quad C = 8, \quad D = 5, \quad E = 1, \quad F = 4, \quad G = 2, \quad H = 7, \quad I = 6$

These values satisfy all the given equations.

Would you like any further details or explanations?

Here are some related questions to expand your understanding:

1. How can systems of linear and nonlinear equations be solved using different methods?
2. What strategies can be used when solving equations with integer constraints?
3. Why is the method of permutations helpful when solving complex puzzles like this one?
4. How can solving such systems of equations be approached computationally versus manually?
5. What are the limitations of using symbolic solutions in complex mathematical systems?

Tip: When solving systems with specific constraints like distinct integers, it can be helpful to reduce the possible search space using logical reasoning before resorting to brute force methods like permutations.