The paper discusses the realization of the ultra-slow roll (USR) phase in the framework of single-field Galileon inflation, which is relevant for explaining the observed scalar-induced gravity waves (SIGWs) from the early universe. The authors examine the generation of SIGWs consistent with pulsar timing array (PTA) data and address the issue of primordial black hole (PBH) overproduction. They incorporate the equation of state (EoS) parameter into their analysis to investigate its impact on PBH formation and SIGW production. The paper outlines the Galileon effective field theory (EFT) setup, including the covariantized Galileon action and the quasi-de Sitter solution. It details the implementation of sharp transitions between slow-roll (SR) and USR phases, analyzing the behavior of slow-roll parameters and EFT coefficients. The authors compute the scalar power spectrum, considering both tree-level and one-loop corrections, and discuss the non-renormalization theorem. They explore the impact of the EoS parameter on PBH formation and SIGW production, presenting numerical outcomes and discussions. The paper concludes with a summary of key findings and their implications for understanding the early universe and the nature of dark matter.The paper discusses the realization of the ultra-slow roll (USR) phase in the framework of single-field Galileon inflation, which is relevant for explaining the observed scalar-induced gravity waves (SIGWs) from the early universe. The authors examine the generation of SIGWs consistent with pulsar timing array (PTA) data and address the issue of primordial black hole (PBH) overproduction. They incorporate the equation of state (EoS) parameter into their analysis to investigate its impact on PBH formation and SIGW production. The paper outlines the Galileon effective field theory (EFT) setup, including the covariantized Galileon action and the quasi-de Sitter solution. It details the implementation of sharp transitions between slow-roll (SR) and USR phases, analyzing the behavior of slow-roll parameters and EFT coefficients. The authors compute the scalar power spectrum, considering both tree-level and one-loop corrections, and discuss the non-renormalization theorem. They explore the impact of the EoS parameter on PBH formation and SIGW production, presenting numerical outcomes and discussions. The paper concludes with a summary of key findings and their implications for understanding the early universe and the nature of dark matter.