Context :

Aims:2015 BZ509 is the first asteroid confirmed to be in retrograde co-orbit resonance ( or 1/-1 resonance ) with the giant planets in the solar system .
While Saturn is the only giant planet whose Trojans are not discovered until now , we identify some small bodies among Centaurs and Damocloids that are potentially in 1/-1 resonance with Saturn in the present study .

Methods : We integrate numerically the motion of the 1000 clones ( include the nominal orbit ) of each Centaur whose orbit has a semi-major axis between 9.3 au and 9.8 au and an inclination i > 90 \degr .
To confirm and evaluate the 1/-1 resonant configurations mentioned above , we introduce an useful one degree integrable approximation for planar 1/-1 resonance .

Results : We identify four candidates potentially in 1/-1 resonance with Saturn .
The capture in this particular resonant state during the 40000 yr integration timespan is very common for 2006 RJ2 ( 906/1000 clones ) , 2006 BZ8 ( 878/1000 clones ) , and 2017 SV13 ( 998/1000 clones ) , and it is less likely for 2012 YE8 ( 426/1000 clones ) .
According to our statistical results , 2006 RJ2 is the best candidate to be currently in a 1/-1 mean motion resonance with Saturn , and 2017 SV13 is another important potential candidate .
Moreover , 2012 YE8 and 2006 BZ8 are also Centaurs of interest but their current and long-term 1/-1 resonant state with Saturn is less likely .
The proportions of the clones captured in the relative long-term stable co-orbit resonance ( over 10000 yr ) are also given .
The motions of the 2006 RJ2 , 2015 BZ509 , and 2006 BZ8 in the solar system are just around the ideal equilibrium points of the 1/-1 resonance given by the planar semi-analytical model .

Conclusions : Small bodies in retrograde co-orbit resonance with giant planets may be more common than previously expected .
Identification of these potential mysterious minor bodies encourages the search for such objects on a larger scale in our solar system .
The findings of this paper are also useful for understanding the origin and dynamical evolution of the Centaurs and Damocloids on retrograde orbits .