  Let x be an element of a group G. For a positive integer n let E_n(x) be the
subgroup generated by all commutators [...[[y,x],x],...,x] over y in G, where x
is repeated n times. There are several recent results showing that certain
properties of groups with small subgroups E_n(x) are close to those of Engel
groups. The present article deals with orderable groups in which, for some n,
the subgroups E_n(x) are polycyclic. Let h,n be positive integers and G an
orderable group in which E_n(x) is polycyclic with Hirsch length at most h for
every x in G. It is proved that there are (h,n)-bounded numbers h* and c* such
that G has a finitely generated normal nilpotent subgroup N with h(N)<h* and
G/N nilpotent of class at most c*.
