Upāṃśu-vrata A term called upāṃśu-vrata can be found in Mahābhārata. It means some serious vow, unknown or previously undisclosed to others. Janamejaya, son of Parīkṣita wished to accept Somaśravā, son of sage Śrutaśravā. The sage proposed that the king should take another of his sons, charged with great ascetic powers, but with a condition. He said, “my son has one upāṃśu-vrata, that is– if a Brāhmaṇ ask for something, he must give that to him. If you can help him in this matter, you can take him along–
asya tu ekaṃ upāṃśuvratam– yadena kaścit brāhmaṇa kañcidartha yācettaṃ tasayai dadyād ayaṃ.

According to Nīlakaṇṭha, upāṃśu-vrata means some serious rule undertaken to be performed very deeply– nigūḍhavanniyamaḥ. And Nīlakaṇṭha has further clarified the meaning of it, saying that upāṃśu vrata is that vrata, the rules and vows of which remain unknown to others. —
upāṃśu-vrata paresāṃ avidita niyamam.

In Droṇaparva of Mahābhārata, in another chapter, it is said that, Arjuna promised to Yudhiṣṭhira that he would kill Karṇa. But in course of adverve situation during the war, Arjuna could not fulfil the promise that far. So Yudhiṣṭhira rebuked him, and asked him give his gāṇḍīva to Kṛṣṇa. Arjuna felt very insulted at these words of the elder brother, and took a khaḍga to slay Yudhiṣṭhira. Seeing that, the great-souled Kṛṣṇa asked him about the matter. Then Arjuna said, I have a secret vow– if anyone asks me to give my gāṇḍīva to another person, I will cut his head off– this is my upāṃśu-vrata.
anyasmai dehi gāṇḍīvamiti māṃ yo̍bhicodayet
bhindyāmahaṃ tasya śira ityupāṃśuvrataṃ mama.

By the intervention of Kṛṣṇa and thanks to his practical advice, the disaster could be avoided, but the secretive nature of this upāṃśu-vrata is notable.