Absolutely! Let's create a test case generator for the given problem.

Given the constraints and the problem statement, the test case generator will need to:
1. Generate a random value for \(N\) within the specified range.
2. Generate two random strings \(S_0\) and \(T\) of length \(N\), consisting of lowercase English letters.

Here's the test case generator for the specified problem:

```python
import random
import string

def generate_random_string(length):
    return ''.join(random.choice(string.ascii_lowercase) for _ in range(length))

def tcgen():
    # Generate N within the range [1, 1,000,000]
    N = random.randint(1, 10**6)
    
    # Generate the string S_0 of length N
    S_0 = generate_random_string(N)
    
    # Generate the string T of length N
    T = generate_random_string(N)
    
    return N, S_0, T

# Example usage
if __name__ == "__main__":
    N, S_0, T = tcgen()
    print(N)
    print(S_0)
    print(T)
```

This script defines a function `generate_random_string` to create a random string of lowercase English letters, and another function `tcgen` that uses it to produce the test case. 

### Explanation:
1. **`generate_random_string` Function**: This function takes a length and returns a random string of that length composed of lowercase English letters.
2. **`tcgen` Function**: 
   - It first generates a random integer \(N\) within the range from 1 to 1,000,000.
   - It then generates two random strings \(S_0\) and \(T\), each of length \(N\).
   - Finally, it returns \(N\), \(S_0\), and \(T\).

### Usage:
When you run this script, it will print the values of \(N\), \(S_0\), and \(T\) in the format required by the problem statement. 

You can integrate this generator into your testing framework to automatically create test cases for your problem.