  We study an XY-rotor model on regular one dimensional lattices by varying the
number of neighbours. The parameter $2\ge\gamma\ge1$ is defined. $\gamma=2$
corresponds to mean field and $\gamma=1$ to nearest neighbours coupling. We
find that for $\gamma<1.5$ the system does not exhibit a phase transition,
while for $\gamma > 1.5$ the mean field second order transition is recovered.
For the critical value $\gamma=\gamma_c=1.5$, the systems can be in a non
trivial fluctuating phase for whichthe magnetisation shows important
fluctuations in a given temperature range, implying an infinite susceptibility.
For all values of $\gamma$ the magnetisation is computed analytically in the
low temperatures range and the magnetised versus non-magnetised state which
depends on the value of $\gamma$ is recovered, confirming the critical value
$\gamma_{c}=1.5$.
