  We develop a method to recognize admissibility of $\Pi_{2}$-rules, relating
this problem to a specific instance of the unification problem with linear
constants restriction, called here "unification with simple variable
restriction". It is shown that for logical systems enjoying an appropriate
algebraic semantics and a finite approximation of left uniform interpolation,
this unification with simple variable restriction can be reduced to standard
unification. As a corollary, we obtain the decidability of admissibility of
$\Pi_{2}$-rules for many logical systems.
