We study the physical properties of the Lyman-alpha forest by applying the inversion method described by Pichon et al. (\cite{Pichon01}) to the high resolution and high S/N ratio spectrum of the z_em=2.40 quasar HE 1122-1628 obtained during Science Verification of UVES at the VLT. We compare the column densities obtained with the new fitting procedure with those derived using standard Voigt profile methods. The agreement is good and gives confidence in the new description of the Lyman-alpha forest as a continuous field as derived from our method. We show that the observed number density of lines with log N > 13 and 14 is, respectively, 50 and 250 per unit redshift at z ~ 2. We study the physical state of the gas, neglecting peculiar velocities, assuming a relation between the overdensity and the temperature, T={/line T}(rho (x)/{bar rho })^2beta . There is an intrinsic degeneracy between the parameters beta and {/line T}. We demonstrate that, at a fixed beta , the temperature at mean density, {/line T}, can be uniquely extracted, however. While applying the method to HE 1122-1628, we conclude that for 0.2 < beta < 0.3, 6000 < {/line T} < 15 000 K at z ~ 2. We investigate the small-scale structure of strong absorption lines using the information derived from the Lyman-beta , Lyman-gamma and C iv profiles. Introducing the Lyman-beta line in the fit allows us to reconstruct the density field up to rho /{bar rho } ~ 10 instead of 5 for the Lyman-alpha line only. The neutral hydrogen density is of the order of ~ 2x 10^-9 cm^-3 and the C IV/H I ratio varies from about 0.001 to 0.01 within the complexes of total column density N(H I) ~ 10¹⁵ cm^-2. Such numbers are expected for photo-ionized gas of density n_H ~ 10^-4 cm^-3 and [C/H] ~ -2.5. There may be small velocity shifts ( ~ 10 km s^-1) between the peaks in the C iv and H I density profiles. Although the statistics is small, it seems that C IV/H I and n_HI are anti-correlated. This could be a consequence of the high sensitivity of the C IV/H I ratio to temperature. The presence of associated O vi absorption, with a similar profile, confirms that the gas is photo-ionized and at a temperature of T ~ 10⁵ K. Based on data collected during Science Verification of the Ultra-violet and Visible Echelle Spectrograph at the European Southern Observatory on the 8.2 m KUEYEN telescope operated on Cerro Paranal, Chile. Table A.1 is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr \ (130.79.128.5) or via \ http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/376/28