  In this work we study the gravitational radiation produced by a keplerian
binary system within the context of Very Special Linear Gravity (VSLG), a novel
theory of linearized gravity in the framework of Very Special Relativity (VSR)
allowing for a gauge-invariant mass $m_g$ of the graviton. For this task, we
exploit Effective Field Theory's techniques, which require, among others, the
calculation of the squared amplitude of the emission process and therefore the
polarization sum for VSLG gravitons. Working in the radiation zone and using
the standard energy momentum tensor's expression for keplerian binaries, we
derive and study the properties of the VSLG energy loss and period decrease
rates, also verifying they reduce to the correct General Relativity limit when
sending $m_g\to0$. Finally, using astronomical data from the Hulse-Taylor
binary and the Double Pulsar J0737-3039, we obtain an upperbound on the VSLG
graviton mass of $m_g\sim 10^{-21}eV$ that, while being comparable to bounds
obtained in this same way for other massive gravity models, is still weaker
than the kinematical bound $\sim 10^{-22}eV$ obtained from the combined
observation of the astronomical events GW170817 and GRB170817A, which should
still hold in VSLG.
