Step 1: Verify Statement 1.
The Election Commission of India (ECI) is established under Article 324 of the Constitution, making it a constitutional body. Statement 1 is true.
Step 2: Verify Statement 2.
ECI conducts elections to Parliament, State Legislatures, and offices of President and Vice President. Elections to Corporations (municipal bodies) are conducted by State Election Commissions, not ECI. Statement 2 is false.
Step 3: Conclusion.
Only Statement 1 is correct.
\[ \boxed{\text{Only the statement (1) is true}} \]