Since we are going to look for the isotropic and homogeneous solution, the starting pointis to choose the metric in the formgμν=a2(η) ¯gμν, whereηis conformal time. It provesuseful to denote, as before,σ= lna. Now, one has to derive the equations for the three fields:ϕ,ψ,andσ. In the rest of this section we shall consider the conformally flat backgroundand thus set ¯gμν=ημν.The equations forϕandψhave especially simple form√−gbracketleftBiggΔ4ϕ+√b8π(E−23✷R)−w8π√bC2bracketrightBigg= 0,√−gbracketleftBiggΔ4ψ−w8π√bC2bracketrightBigg= 0.One has to remind the transformation law for the quantities which enter the last expression:√−gC2=√−¯g¯C2,√−gΔ4=√−¯g¯Δ4,(11)√−g(E−23✷R) =√−¯g(¯E−23¯✷¯R+ 4¯Δ4σ).(12)Taking into account our choice for the fiducial metric ¯gμν=ημν, one arrives at the followingequations in flat space-time✷2ϕ+√b2π✷2σ= 0,✷2ψ= 0.(13)The solutions of (13) can be presented in the formϕ=−√b2πσ+ϕ0,ψ=ψ0.(14)whereϕ0,ψ0are general solutions of the homogeneous equations✷2ϕ0= 0,✷2ψ0= 0.Thus one meets an arbitrariness related to the choice of the initial conditions for the auxiliaryfieldsϕ,ψ. But, the inflationary solution does not depend onϕ0,ψ0. Substituting (14) backinto the action and taking variation with respect toσwe arrive at the same equation forσthat follows directly from (6). We shall write this equation in terms of the physical timet,defined, as usual, througha(η)dη=dt. The useful variable isH(t) = ˙a(t)/a(t) = ˙σ(t), sincethe equation (completely equivalent to the one of [7]) is of the third order in this variable:...H+ 7..HH+ 4parenleftBigg1 +3bcparenrightBigg.HH2+ 4.H2+4bcH4−2M2PcparenleftBigH2+.HparenrightBig= 0.(15)It is easy to identify the special solution corresponding toH=const:H=±MP√b,a(t) =a0·expHt.(16)8