  In the last two decades the study of random instances of constraint
satisfaction problems (CSPs) has flourished across several disciplines,
including computer science, mathematics and physics. The diversity of the
developed methods, on the rigorous and non-rigorous side, has led to major
advances regarding both the theoretical as well as the applied viewpoints.
Based on a ceteris paribus approach in terms of the density evolution equations
known from statistical physics, we focus on a specific prominent class of
regular CSPs, the so-called occupation problems. The regular $r$-in-$k$
occupation problems resemble a basis of this class. By now, out of these CSPs
only the satisfiability threshold - the largest degree for which the problem
admits asymptotically a solution - for the $1$-in-$k$ occupation problem has
been rigorously established. Here we determine the satisfiability threshold of
the $2$-in-$k$ occupation problem for all $k$. In the proof we exploit the
connection of an associated optimization problem regarding the overlap of
satisfying assignements to a fixed point problem inspired by belief
propagation, a message passing algorithm developed for solving such CSPs.
