Some two color, four variable Rado numbers

There exists a minimum integer N such that any 2-coloring of { 1 , 2 , … , N } admits a monochromatic solution to x + y + k z = ℓ w for k , ℓ ∈ Z + , where N depends on k and ℓ. We determine N when ℓ − k ∈ { 0 , 1 , 2 , 3 , 4 , 5 } , for all k , ℓ for which 1 2 ( ( ℓ − k ) 2 − 2 ) ( ℓ − k + 1 ) ⩽ k ⩽ ℓ − 4 , as well as for arbitrary k when ℓ = 2 .