  In the present paper, I have re-examined the weak basis invariants at low
energies, proposed by C. Jarlskog and Branco et. al, respectively, in their
earlier analyses, after confronting them with the assumptions of two zeros and
an equality between arbitrary non-zero elements in the Majorana neutrino mass
matrix in the flavoured basis. This particular conjecture is found to be
experimentally feasible as shown by S. Dev and D. Raj in their recent work. The
present analysis attempts to find the necessary and sufficient condition for CP
invariance for each experimentally viable ans\"atz, pertaining to the model
along with some important implications.
