From $\lambda$-connections to $PS{L}_2(\mathbb{C})$-opers with apparent singularities

On a Riemann surface of genus $>1$, we discuss how to construct opers with apparent singularities from $S{L}_2(\mathbb{C})$ $\lambda$-connections $(E,{\mathrm{\nabla }}_{\lambda })$ and sub-line bundles $L$ of $E$. This construction defines a rational map from a space which captures important data of triples $(E,L,{\mathrm{\nabla }}_{\lambda })$ 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.