  In the course of a project to study eclipsing binary stars in vinicity of the
Sun, we found that the cooler component of LL Aqr is a solar twin candidate.
This is the first known star with properties of a solar twin existing in a
non-interacting eclipsing binary, offering an excellent opportunity to fully
characterise its physical properties with very high precision. We used
extensive multi-band, archival photometry and the Super-WASP project and
high-resolution spectroscopy obtained from the HARPS and CORALIE spectrographs.
The spectra of both components were decomposed and a detailed LTE abundance
analysis was performed. The light and radial velocity curves were simultanously
analysed with the Wilson-Devinney code. The resulting highly precise stellar
parameters were used for a detailed comparison with PARSEC, MESA, and GARSTEC
stellar evolution models. LL Aqr consists of two main-sequence stars (F9 V + G3
V) with masses of M1 = 1.1949$\pm$0.0007 and M2=1.0337$\pm$0.0007 $M_\odot$,
radii R1 = 1.321$\pm$0.006 and R2 = 1.002$\pm$0.005 $R_\odot$, temperatures
T1=6080$\pm$45 K and T2=5703$\pm$50 K and solar chemical composition
[M/H]=0.02$\pm$0.05 dex. The absolute dimensions, radiative and photometric
properties, and atmospheric abundances of the secondary are all fully
consistent with being a solar twin. Both stars are cooler by about 3.5 $\sigma$
or less metal abundant by 5$\sigma$ than predicted by standard sets of stellar
evolution models. When advanced modelling was performed, we found that full
agreement with observations can only be obtained for values of the mixing
length and envelope overshooting parameters that are hard to accept. The most
reasonable and physically justified model fits found with MESA and GARSTEC
codes still have discrepancies with observations but only at the level of
1$\sigma$.
