JuliaSimCompiler fully simplifies the system:
julia> structural_simplify(IRSystem(fols))
States (0):
Variables (2):
1 => y
2 => x
Matched SystemStructure with 2 equations and 2 variables
# ∂ₜ eq # ∂ₜ v
1 [(1), 2] | 1 [(1), 2]
2 [1, 2] | 2 [1, 2]
Legend: Solvable | (Solvable + Matched) | Unsolvable | (Unsolvable + Matched) | ∫ SelectedState