Sure, here is a Python test case generator for the given problem statement:

```python
import random

def tcgen():
    N = random.randint(1, 50)

    balls = []
    x_coords = set()
    y_coords = set()

    for _ in range(N):
        while True:
            x = random.randint(-10**9, 10**9)
            y = random.randint(-10**9, 10**9)
            # Ensure unique x or y coordinates as per problem constraints
            if x not in x_coords and y not in y_coords:
                x_coords.add(x)
                y_coords.add(y)
                balls.append((x, y))
                break

    return N, balls

# Example usage
N, balls = tcgen()
print(N)
for x, y in balls:
    print(x, y)
```

This generator ensures that:
1. `N` is chosen randomly between 1 and 50.
2. Each ball's coordinates `(x, y)` are chosen such that `|x|` and `|y|` each can be as large as \(10^9\).
3. Coordinates of each ball are unique in terms of either `x` or `y` (ensuring `x_i != x_j` or `y_i != y_j` for `i != j`).

You can use the `tcgen` function to generate test cases for your problem. Each call to `tcgen` will produce a new set of test data adhering to the constraints specified.