Monodromy of the fibre with oscillatory singular point of type $1:(-2)$

In the present work, we prove the existence of fractional monodromy in a large class of compact Lagrangian fibrations of four-dimensional symplectic manifolds. These fibrations are considered in the neighbourhood of the singular fibre $\lambda_0$, that has a single singular point corresponding to a nonlinear oscillator with frequencies in $1:(-2)$ resonance. We compute the matrices of monodromy defined by going around the fibre $\lambda_0$. For all fibrations in the class and for an appropriate choice of the basis in the one-dimensional homology group of the torus, these matrices are the same. The elements of the monodromy matrix are rational and there is a non-integer element among them. This work is a continuation of the analysis in [20, 21, 39] where the matrix of fractional monodromy was computed for most simple particular fibrations of the class.