Infinite-dimensional linear groups with restrictions on subgroups that are not soluble $A_3$-groups

We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble $A_3$-groups and all of whose proper subgroups, which are not soluble $A_3$-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble $A_3$-groups and all of whose proper subgroups, which are not soluble $A_3$-groups, have finite fundamental dimension.