Helmholtz theorem and the v-gauge in the problem of superluminal and   instantaneous signals in classical electrodynamics
  In this work we substantiate the applying of the Helmholtz vector decomposition theorem (H-theorem) to vector fields in classical electrodynamics. Using the H-theorem, within the framework of the two-parameter Lorentz-like gauge (so called v-gauge), we show that two kinds of magnetic vector potentials exist: one of them (solenoidal) can act exclusively with the velocity of light c and the other one (irrotational) with an arbitrary finite velocity $v$ (including a velocity more than c . We show also that the irrotational component of the electric field has a physical meaning and can propagate exclusively instantaneously. 
Andrew Chubykalo, Augusto Espinoza, Alvarado-Flores and Alejandro   Gutierrez Rodriguez