  Motivated by a classic result for free groups, one says that a group $G$ has
the Magnus property if the following holds: whenever two elements generate the
same normal subgroup of $G$, they are conjugate or inverse-conjugate in $G$.
  It is a natural problem to find out which relatively free groups display the
Magnus property. We prove that a free polynilpotent group of any given class
row has the Magnus property if and only if it is nilpotent of class at most
$2$. For this purpose we explore the Magnus property more generally in soluble
groups, and we produce new techniques, both for establishing and for disproving
the property. We also prove that a free centre-by-(polynilpotent of given class
row) group has the Magnus property if and only if it is nilpotent of class at
most $2$.
  On the way, we display $2$-generated nilpotent groups (with non-trivial
torsion) of any prescribed nilpotency class with the Magnus property. Similar
examples of finitely generated, torsion-free nilpotent groups are hard to come
by, but we construct a $4$-generated, torsion-free, class-$3$ nilpotent group
of Hirsch length $9$ with the Magnus property. Furthermore, using a weak
variant of the Magnus property and an ultraproduct construction, we establish
the existence of metabelian, torsion-free, nilpotent groups of any prescribed
nilpotency class with the Magnus property.
