Sure, I can help you write a test case generator for the "Soccer" problem statement. Here is a Python function that generates random test cases based on the given constraints:

```python
import random

def generate_test_case():
    # Field dimensions
    H = random.randint(1, 500)
    W = random.randint(1, 500)
    
    # Fatigue degree increments
    A = random.randint(0, 10**9)
    B = random.randint(0, 10**9)
    C = random.randint(0, 10**9)
    
    # Number of players
    N = random.randint(2, 100000)
    
    # Player positions
    S = [random.randint(0, H) for _ in range(N)]
    T = [random.randint(0, W) for _ in range(N)]
    
    # Make sure player 1 (index 0) and player N (index N-1) are not at the same position
    while S[0] == S[N-1] and T[0] == T[N-1]:
        S[N-1] = random.randint(0, H)
        T[N-1] = random.randint(0, W)
    
    # Formatting the test case as required by the problem statement
    test_case = f"{H} {W}\n"
    test_case += f"{A} {B} {C}\n"
    test_case += f"{N}\n"
    
    for i in range(N):
        test_case += f"{S[i]} {T[i]}\n"
    
    return test_case

# Example usage:
print(generate_test_case())
```

This function generates a random test case according to the problem constraints:

1. Randomly sets the height `H` and width `W` of the field.
2. Randomly determines the fatigue degree increments `A`, `B`, and `C`.
3. Randomly generates the number of players `N`.
4. Randomly assigns initial positions `(S_i, T_i)` for each player, ensuring that the first player (with the ball) and the last player (where the ball needs to be passed to) are not at the same position.

You can call `generate_test_case()` to get a new random test case each time. Adjust the `print` statement as needed to integrate with your testing framework.