Optical observations of the GRB 970508 optical remnant were continued with the 6-m telescope of SAO RAS in standard BVRcIc bands in Oct.-Dec.1997 and in Jan. 1998. The results of the photometry of GRB970508 remnant and of three nearby galaxies are presented. The BVRcIc light curves of the GRB970508 remnant may be described by a power law plus a constant F(t)=F_0(t-t_0)^{alpha}+C. In In parameters determination of the faint host galaxy we used the results of our BVRcIc photometry of May-August, 1997, the data of recent observations Keck-II and WHT telescopes and also the data of Rc and B bands photometry obtained by other authors based on our secondary standards. The level-off from the initial power-law decline seen in the first months after the burst observed in all bands. The effect is the most strong in Ic band. The best chi^2-fits for Fo, alpha, C parameters of the data in each of 4 bands point to the presence of a faint constant source with Ic=24.13pm0.28, Rc = 25.55pm0.19, V=25.80pm 0.14, B=26.68pm 0.14. The average alpha of light curve power law is -1.23pm0.04. The optical remnant has the power law F_ u u^beta spectrum with spectral slope = -1.10pm0.08 and does not change after optical curve maximum. BVRcIc spectrum together with an absolute magnitude of the constant component M_B_rest=-17.5pm0.3 and the limit of the linear size of underlying host galaxy d<1.1 kpc, calculated for z=0.835,H_0=75km s^-1 Mpc^-1, q_0=0.5, correspond to a host galaxy, such as a starburst dwarf, red starburst dwarf, irregular dwarf, HII galaxy, or blue compact dwarf galaxy. All these types of dwarf galaxies show evidence of starburst or post starburst activity. Galaxy G2 has a spectrum similar to one of the host GRB galaxy and lies at the projected distance of ~20 kpc from GRB.
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