Existence-bound-word subgroup

Symbol-free definition
A subgroup of a group is termed an existence-bound-word subgroup if there exists a collection of equations, each in one unknown and several parameters, such that as the parameters vary over elements of the group, the set of possible solutions to each equation, together, form a generating set for the subgroup.

Definition with symbols
Let $$E$$ be a collection of equations, each equation featuring one unknown and a finite number of parameter variables. For each equation, define the possible solution set to be the set of possible values of the unknown variable for which the equation has a solution. Consider the subgroup generated by the union of possible solution sets for each equation in $$E$$. A subgroup obtained in this way is called an existence-bound-word subgroup.

Stronger properties

 * Verbal subgroup

Weaker properties

 * Fully characteristic subgroup
 * Bound-word subgroup