Mumbai, Jan. 29 -- Indian Rupee stabilized a bit after sharp losses in last session as market focused on the latest regulatory attempts after the local currency fell to record lows against the US dollar over last few days. As part of its over Rs 1.5 lakh crore liquidity injection into the banking system, the Reserve Bank on Tuesday said the Reserve Bank will be conducting a USD/INR Buy/Sell swap auction of USD 5 billion on January 31 for a tenor of 6 months. INR saw an overall muted movement following this and currently trades flat around 86.57 per US dollar. While this is still near record low mark, the INR has somewhat managed to hold on its own despite firm cues from the US dollar index which quotes at 107.85, up 0.15% on the day and ris...