Each person should recite the Vazhi Thirunamam of their own sampradaya. It is not that one should not recite the Vazhi Thirunamam of other traditions; any of them can be recited. However, one should recite the Vazhi Thirunamam that aligns with their respective sampradaya. There is no such thing as a separate Thennacharya Vazhi Thirunamam. They recite the one belonging to Manavala Mamunigal. That is not recited in the Shrimad Ahobila Mutt. Even within the Thennacharya sampradaya, they do not recite the Vazhi Thirunamams of those like Nanjiyar, Nampillai, or Periyavachan Pillai. What is recited in Shrimad Ahobila Mutt is not the Thennacharya Vazhi Thirunamam; it is our Acharya Vazhi Thirunamam. Since these are universal Acharyas, you may recite them freely. Morals Revealed by Sampradaya

