A problem of existence and uniqueness of the solution and its approximate construction for the system of differential equations with a vector of slow variables a ∈ Δ ⊂ Rn and fast variables φ ∈ Tm of the form da dτ = X(τ, aΛ,φΘ), dφ dτ = ω(τ ) ε + Y (τ, aΛ,φΘ), (1) are investigated in the paper. Here τ ∈ [0,L], small parameter ε ∈ (0, ε0], 0 < λ1 < · · · < λp ≤ 1, 0 < θ1 < · · · < θq ≤ 1, aΛ = (aλ1, . . . ,aλp ), aλi (τ) = a(λiτ ), φΘ = (φθ1, . . . ,φθq ), φθj (τ) = φ(θjτ ). Multifrequency ODE systems are researched in detail in [1], systems with delayed argument were studied in [2, 3], etc. Conditions are set for the system (1) r ν=1 ανa(τν) = s ν=1 ην ξν fν(τ, aΛ,φΘ)dτ, r ν=1 βνφ(τν) = s ν=1 ην ξν gν(τ, aΛ,φΘ)dτ, (2) where [ξν, ην] ⊂ [0,L], ∩ν [ξν, ην] = ∅. For the problem (1), (2) a much simpler problem is constructed by averaging over fast variables φθν on the cube of periods [0, 2π]mq. A sufficient condition for the system of equations (1) to exit a small circumference of the resonance of frequencies ω(τ ) was found, the condition of which in the point τ ∈ [0,L] is q ν=1 θν  kν,ω(θντ )  = 0,kν ∈ Zm, ∥k1∥ + · · · + ∥kq∥ ̸= 0. It is proved that for the smooth enough right parts of the system (1) and sub-integral functions under conditions (2), the condition to exit the circumference of resonances and for small enough ε0 > 0 there exists a unique solution for the problem (1), (2) and an estimation is found ∥a(τ ; y,ψ, ε)−a(τ ; y)∥+∥φ(τ ; y,ψ, ε)−φ(τ ; y,ψ, ε)∥ ≤ c1εα,α = (mq)−1, where c1 > 0 and does not depend on ε,  a(0; y,ψ, ε),φ(0; y,ψ, ε)  = (y,ψ),  a(τ ; y), φ(τ ; y,ψ, ε)  – the solution of the averaged problem with initial conditions (y, ψ), while ∥y − y∥ + ∥ψ − ψ∥ ≤ c2εα.