We present our first results on a new sample containing all single G, K and M giants down to V = 4 mag in the Solar vicinity, suitable for spectropolarimetric (Stokes V) observations with Narval at TBL, France. For detection and measurement of the magnetic field (MF), the Least Squares Deconvolution (LSD) method was applied (Donati et al. 1997) that in the present case enables detection of large-scale MFs even weaker than the solar one (the typical precision of our longitudinal MF measurements is 0.1-0.2 G). The evolutionary status of the stars is determined on the basis of the evolutionary models with rotation (Lagarde et al. 2012; Charbonnel et al., in prep.) and fundamental parameters given by Massarotti et al. (1998). The stars appear to be in the mass range 1-4 M⊙, situated at different evolutionary stages after the Main Sequence (MS), up to the Asymptotic Giant Branch (AGB).
The sample contains 45 stars. Up to now, 29 stars are observed (that is about 64% of the sample), each observed at least twice. For 2 stars in the Hertzsprung gap, one is definitely Zeeman detected. Only 5 G and K giants, situated mainly at the base of the Red Giant Branch (RGB) and in the He-burning phase are detected. Surprisingly, a lot of stars ascending towards the RGB tip and in early AGB phase are detected (8 of 13 observed stars). For all Zeeman detected stars v sin i is redetermined and appears in the interval 2-3 km/s, but few giants with MF possess larger v sin i.