High-resolution (lambda/d lambda similar to 41 000) echelle spectra of the multiple system eta Muscae were analysed. Precise spectroscopic orbital elements of the close pair eta Muscae-A were obtained by means of two techniques; cross-correlation and spectral disentangling. The updated light elements of the close pair were derived using new photometric and spectroscopic data together with Hipparcos ephemerides. According to simultaneous studies of radial velocities with Hipparcos light curve and recent and historic light curves in Stromgren bands, eta Mus-A consists of two quasi-identical components: M(1) =3.30 M., M(2) = 3.29 M., R(1) = 2.14 R., R(2) = 2.13 R., T(1)=12 700 K and T(2) = 12 550 K separated from each other by a = 14.11 R. in an orbit inclined 77 degrees.4. The membership of eta Mus-A to the Lower Centaurus-Crux OB association is discussed. The rotational velocity of the primary component was found to be smaller than that of the secondary component. The asynchronous slow rotation of the primary and the synchronous rotation of the secondary imply a recent decrease in the orbital period, either due to orbital angular momentum being transferred to the visual companion eta Mus-C or due to relatively frequent stellar encounters in this crowded region. The visual companion eta Mus-B is found to be gravitationally unbound to the close pair eta Mus-A. The radial velocity measurements of the visual companion eta Mus-B, when combined with its positional information, support its membership to the Lower Centaurus-Crux OB association, but the evidence for membership of eta Mus-A is weaker.