Unlike the first two families, this solution imposesno dispersion relation constraintω 2 =k 2c2

Pure Gauge Solution The parameters satisfy the following conditions: α1 =η ω 2gc , α 2 =η k 2g , α 3 =− λ 2g , α 5 =ηα 4, η=±1, α 4 ̸= 0, α 1, α 2 ̸= 0, λ∈R