  By a theorem of Mandell-May-Schwede-Shipley the stable homotopy theory of
classical $S^1$-spectra is recovered from orthogonal spectra. In this paper
general linear, special linear, symplectic, orthogonal and special orthogonal
motivic spectra are introduced and studied. It is shown that the stable
homotopy theory of motivic spectra is recovered from each of these types of
spectra. An application is given for the localization functor $C_*\mathcal
Fr:SH_{nis}(k)\to SH_{nis}(k)$ in the sense of [15] that converts the
Morel-Voevodsky stable motivic homotopy theory $SH(k)$ into the equivalent
local theory of framed bispectra [15].
