Monodromy Equivalence for Lamé-type Equations I: Finite-gap Structures and Cone Spherical Metrics

Motivated by the finite-gap structure of the classical Lamé equation (1.2) and its central role in mathematical physics, generalized Lamé-type equations (1.12) are investigated. For the fundamental case $n=1$, a monodromy equivalence between the classical Lamé equation (1.18) and the generalized Lamé-type equation (1.19) is established. Two main applications are obtained: (i) the finite-gap structure of \ (1.19) is derived, together with a complete classification of the spectral curves $σ_{1}$ and $σ_{2}$ for $τ\in i\mathbb{R}_{>0}$; and (ii) the monodromy equivalence is applied to the construction of cone spherical metrics with three large conical singularities, each with cone angle exceeding $2π$. A family of such metrics is shown to exhibits a blow-up configuration, which is described explicitly in terms of the monodromy data.