From $\lambda$-connections to $PSL_2(\mathbb{C})$-opers with apparent singularities

On a Riemann surface of genus $> 1$, we discuss how to construct opers with apparent singularities from $SL_2(\mathbb{C})$ $\lambda$-connections $(E, \nabla_\lambda)$ and sub-line bundles $L$ of $E$. This construction defines a rational map from a space which captures important data of triples $(E, L, \nabla_\lambda)$ to a space which parametrises the positions and residue parameters of the induced apparent singularities. We show that this is a Poisson map with respect to natural Poisson structures. The relations to wobbly bundles and Lagrangians in the moduli spaces of Higgs bundles and $\lambda$-connections are discussed.