  We aim to detect the magnetic field and infer its properties around four AGB
stars using H$_2$O maser observations. The sample we observed consists of the
following sources: the semi-regular variable RT Vir and the Mira variables AP
Lyn, IK Tau, and IRC+60370. We observed the 6$_{1,6}-5_{2,3}$ H$_2$O maser
rotational transition, in full-polarization mode, to determine its linear and
circular polarization. Based on the Zeeman effect, one can infer the properties
of the magnetic field from the maser polarization analysis. We detected a total
of 238 maser features, in three of the four observed sources. No masers were
found toward AP Lyn. The observed masers are all located between 2.4 and 53.0
AU from the stars. Linear and circular polarization was found in 18 and 11
maser features, respectively. We more than doubled the number of AGB stars in
which magnetic field has been detected from H$_2$O maser polarization, as our
results confirm the presence of fields around IK Tau, RT Vir and IRC+60370. The
strength of the field along the line of sight is found to be between 47 and 331
mG in the H$_2$O maser region. Extrapolating this result to the surface of the
stars, assuming a toroidal field ($\propto$ r$^{-1}$), we find magnetic fields
of 0.3-6.9 G on the stellar surfaces. If, instead of a toroidal field, we
assume a poloidal field ($\propto$ r$^{-2}$), then the extrapolated magnetic
field strength on the stellar surfaces are in the range between 2.2 and
$\sim$115 G. Finally, if a dipole field ($\propto$ r$^{-3}$) is assumed, the
field strength on the surface of the star is found to be between 15.8 and
$\sim$1945 G. The magnetic energy of our sources is higher than the thermal and
kinetic energy in the H$_2$O maser region of this class of objects. This leads
us to conclude that, indeed, magnetic fields probably play an important role in
shaping the outflows of evolved stars. (abridged)
