Construction of a possible Lagrangian for the ODE's system

Question

Given:

$\begin{cases} x''-(x-(y-z))=0 \\ y''+(x-(y-z))=0 \\ z''-f''y-f'y'=0 \end{cases}$

$x,y,z$ - variables, $f$ - arbitrary time-dependent function

The Helmholtz conditions for this system are not all satisfied. It is difficult to solve the system of Helmholtz equations because of the large dimension. Is there an alternative way to construct a possible Lagrangian for this system of equations?

eqns = {Subscript[c, 
     12] (Subscript[\[Phi], 1][t] - Subscript[\[Phi], 2][t] + 
       Subscript[\[CapitalPhi], 2][t]) + 
    Subscript[J, 
     1] (Subscript[\[Phi], 1]^\[Prime]\[Prime])[t], -Subscript[c, 
      12] (Subscript[\[Phi], 1][t] - Subscript[\[Phi], 2][t] + 
       Subscript[\[CapitalPhi], 2][t]) + 
    Subscript[J, 2] (Subscript[\[Phi], 2]^\[Prime]\[Prime])[t], 
   Subscript[\[CapitalPhi], 2]''[t] - A''[t] Subscript[\[Phi], 2][t] -
     A'[t] Subscript[\[Phi], 2]'[t]};

g1 = {{D[eqns[[1]], Subscript[\[Phi], 1][t]], 
     D[eqns[[1]], Subscript[\[Phi], 1]'[t]], 
     D[eqns[[1]], Subscript[\[Phi], 1]''[t]]}, {D[eqns[[2]], 
      Subscript[\[Phi], 1][t]], 
     D[eqns[[2]], Subscript[\[Phi], 1]'[t]], 
     D[eqns[[2]], Subscript[\[Phi], 1]''[t]]}, {D[eqns[[3]], 
      Subscript[\[Phi], 1][t]], 
     D[eqns[[3]], Subscript[\[Phi], 1]'[t]], 
     D[eqns[[3]], Subscript[\[Phi], 1]''[t]]}} // MatrixForm;

g2 = {{D[eqns[[1]], Subscript[\[Phi], 2][t]], 
     D[eqns[[1]], Subscript[\[Phi], 2]'[t]], 
     D[eqns[[1]], Subscript[\[Phi], 2]''[t]]}, {D[eqns[[2]], 
      Subscript[\[Phi], 2][t]], 
     D[eqns[[2]], Subscript[\[Phi], 2]'[t]], 
     D[eqns[[2]], Subscript[\[Phi], 2]''[t]]}, {D[eqns[[3]], 
      Subscript[\[Phi], 2][t]], 
     D[eqns[[3]], Subscript[\[Phi], 2]'[t]], 
     D[eqns[[3]], Subscript[\[Phi], 2]''[t]]}} // MatrixForm;

g3 = {{D[eqns[[1]], Subscript[\[CapitalPhi], 2][t]], 
     D[eqns[[1]], Subscript[\[CapitalPhi], 2]'[t]], 
     D[eqns[[1]], Subscript[\[CapitalPhi], 2]''[t]]}, {D[eqns[[2]], 
      Subscript[\[CapitalPhi], 2][t]], 
     D[eqns[[2]], Subscript[\[CapitalPhi], 2]'[t]], 
     D[eqns[[2]], Subscript[\[CapitalPhi], 2]''[t]]}, {D[eqns[[3]], 
      Subscript[\[CapitalPhi], 2][t]], 
     D[eqns[[3]], Subscript[\[CapitalPhi], 2]'[t]], 
     D[eqns[[3]], Subscript[\[CapitalPhi], 2]''[t]]}} // MatrixForm;

Links I have used:

1. Inverse problem for Lagrangian mechanics

2. A Brief Review of Helmholtz Conditions

3. Helmholtz conditions and the inverse problem for Lagrangian mechanics. P. 13