Sure, let's write a test case generator for the given problem statement.

The task is to generate test cases for the problem, which involves generating random strings \( S_0 \) and \( T \) of a given length \( N \). Here’s a Python function to generate such test cases:

```python
import random
import string

def tcgen():
    # Step 1: Generate a random N within the given constraints
    N = random.randint(1, 10**6)
    
    # Step 2: Generate a random string S_0 of length N
    S_0 = ''.join(random.choices(string.ascii_lowercase, k=N))
    
    # Step 3: Generate a random string T of length N
    T = ''.join(random.choices(string.ascii_lowercase, k=N))
    
    return N, S_0, T

# Example usage:
N, S_0, T = tcgen()
print(N)
print(S_0)
print(T)
```

Explanation:

1. **Step 1**: We generate a random integer \( N \) within the range [1, 1,000,000] using `random.randint`.

2. **Step 2**: We generate a random string \( S_0 \) of length \( N \) using `random.choices` with `string.ascii_lowercase` to get lowercase English letters.

3. **Step 3**: We generate another random string \( T \) of length \( N \), following the same approach as for \( S_0 \).

The function `tcgen()` will return a tuple containing the generated values \( N \), \( S_0 \), and \( T \).

This generator can be used to create various test cases for competitive programming problems by calling the function and printing or using its output as needed.