  Let $X$ be a regular one-dimensional transient diffusion and $L^y$ be its
local time at $y$. The stochastic differential equation (SDE) whose solution
corresponds to the process $X$ conditioned on $[L^y_{\infty}=a]$ for a given
$a\geq 0$ is constructed and a new path decomposition result for transient
diffusions is given. In the course of the construction of the SDE the concept
of {\em recurrent transformation} is introduced and {\em Bessel-type motions}
as well as their SDE representations are studied. A remarkable link between an
$h$-transform with a minimal excessive function and recurrent transformations
is found, which, as a by-product, gives a useful representation of last passage
times as a mixture of first hitting times. Moreover, the Engelbert-Schmidt
theory for the weak solutions of one dimensional SDEs is extended to the case
when the initial condition is an entrance boundary for the diffusion. This
extension was necessary for the construction of the Bessel-type motion which
played an essential part in the SDE representation of $X$ conditioned on
$[L^y_{\infty}=a]$.
