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In the Mathur-Mukhi-Sen (MMS) classification scheme for rational conformalfield theories (RCFTs), a RCFT is identified by a pair of non-negative integers$\mathbf{[n, \ell]}$, with $\mathbf{n}$ being the number of characters and$\mathbf{\ell}$ the Wronskian index. The modular linear differential equation(MLDE) that the characters of a RCFT solve are labelled similarly. All RCFTswith a given $\mathbf{[n, \ell]}$ solve the modular linear differentialequation (MLDE) labelled by the same $\mathbf{[n, \ell]}$. With the goal ofclassifying $\mathbf{[3,3]}$ and $\mathbf{[3,4]}$ CFTs, we set-up and solvethose MLDEs, each of which is a three-parameter non-rigid MLDE, forcharacter-like solutions. In the former case, we obtain four infinite familiesand a discrete set of $15$ solutions, all in the range $0 < c \leq 32$. Amongstthese $\mathbf{[3,3]}$ character-like solutions, we find pairs of them thatform coset-bilinear relations with meromorphic CFTs/characters of centralcharges $16, 24, 32, 40, 48, 56, 64$. There are six families of coset-bilinearrelations where both the RCFTs of the pair are drawn from the infinite familiesof solutions. There are an additional $23$ coset-bilinear relations betweencharacter-like solutions of the discrete set. The coset-bilinear relationsshould help in identifying the $\mathbf{[3,3]}$ CFTs. In the $\mathbf{[3,4]}$case, we obtain nine character-like solutions each of which is a$\mathbf{[2,2]}$ character-like solution adjoined with a constant character.

Classifying three-character RCFTs with Wronskian index equalling 3 or 4