  Starting from an anti-symplectic involution on a K3 surface, one can consider
a natural Lagrangian subvariety inside the moduli space of sheaves over the K3.
One can also construct a Prymian integrable system following a construction of
Markushevich--Tikhomirov, extended by Arbarello--Sacc\`a--Ferretti, Matteini
and Sawon--Chen. In this article we address a question of Sawon, showing that
these integrable systems and their associated natural Lagrangians degenerate,
respectively, into fix loci of involutions considered by Heller--Schaposnik,
Garcia-Prada--Wilkins and Basu--Garcia-Prada.
  Along the way we find interesting results such as the proof that the
Donagi--Ein--Lazarsfeled degeneration is a degeneration of symplectic
varieties, a generalization of this degeneration, originally described for K3
surfaces, to the case of an arbitrary smooth projective surface, and a
description of the behaviour of certain involutions under this degeneration.
