  In this paper nontrivial Killing vector fields of constant length and
corresponding flows on smooth complete Riemannian manifolds are investigated.
It is proved that such a flow on symmetric space is free or induced by a free
isometric action of the circle $S^1$. The properties of the set of all points
with finite (infinite) period for general isometric flow on Riemannian
manifolds are described. It is shown that this flow is generated by an
effective almost free isometric action of the group $S^1$ if there are no
points of infinite or zero period. In the last case the set of periods is at
most countable and naturally generates an invariant stratification with closed
totally geodesic strata; the union of all regular orbits is open connected
everywhere dense subset of complete measure. Examples of unit Killing vector
fields generated by almost free but not free actions of $S^1$ on Riemannian
manifolds close in some sense to symmetric spaces are constructed; among them
are "almost round" odd-dimensional spheres, homogeneous (non simply connected)
Riemannian manifolds of constant positive sectional curvature, locally
Euclidean spaces, and unit vector bundles over Riemannian manifolds. Some
curvature restrictions on Riemannian manifolds admitting nontrivial Killing
vector fields of constant length are obtained. Some unsolved questions are
formulated.
