  From a spray space $S$ on a manifold $M$ we construct a new geometric space
$P$ of larger dimension with the following properties:
  1. Geodesics in $P$ are in one-to-one correspondence with parallel Jacobi
fields of $M$.
  2. $P$ is complete if and only if $S$ is complete.
  3. If two geodesics in $P$ meet at one point, the geodesics coincide on their
common domain, and $P$ has no conjugate points.
  4. There exists a submersion $\pi\colon P \to M$ that maps geodesics in $P$
into geodesics on $M$.
  Space $P$ is constructed by first taking two complete lifts of spray $S$.
This will give a spray $S^{cc}$ on the second iterated tangent bundle $TTM$.
Then space $P$ is obtained by restricting tangent vectors of geodesics for
$S^{cc}$ onto a suitable $(2\dim M+2)$-dimensional submanifold of $TTTM$. Due
to the last restriction, space $P$ is not a spray space. However, the
construction shows that conjugate points can be removed if we add dimensions
and relax assumptions on the geometric structure.
