  Gale and Shapley's stable assignment problem has been extensively studied,
applied, and extended. In the context of school choice, mechanisms often aim at
finding an assignment that is more favorable to students. We investigate two
extensions introduced in this framework -- legal assignments and the EADAM
algorithm -- through the lens of classical theory of stable matchings. In any
instance, the set ${\cal L}$ of legal assignments is known to contain all
stable assignments. We prove that ${\cal L}$ is exactly the set of stable
assignments in another instance. Moreover, we show that essentially all
optimization problems over ${\cal L}$ can be solved within the same time bound
needed for solving it over the set of stable assignments. A key tool for this
latter result is an algorithm that finds the student-optimal legal assignment.
We then generalize our algorithm to obtain the assignment output of EADAM with
any given set of consenting students without sacrificing the running time,
hence largely improving in both theory and practice over known algorithms.
Lastly, we show that the set ${\cal L}$ can be much larger than the set of
stable matchings, connecting legal matchings with certain concepts and open
problems in the literature.
