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The unusually low amplitude anisotropic wave train events (LAWEs) in cosmic ray intensity using the ground based Deep River neutron monitor data has been studied during the period 1991–1994. It has been observed that the amplitude of the diurnal anisotropy for LAWE events significantly remains quite low and statistically constant as compared to the quiet day annual average amplitude for majority of the events. The time of maximum of the diurnal anisotropy of LAWE significantly shifts towards earlier hours as compared to the co-rotational direction and remains in the direction of quiet day annual average anisotropy for majority of the events. On the other hand, the amplitude of the semi/tri-diurnal anisotropy remains statistically the same and high whereas, phase shift towards later hours as compared to the quiet day annual average values for majority of the LAWEs. The diurnal anisotropy vectors are found to shifts towards earlier hours for 50% of the events; whereas they are found to shifts towards later hours for rest of the events (50%) relative to the average vector for the entire period. It is also noted that the amplitude of these vectors are found to increase significantly with the shift of the diurnal anisotropy vectors towards later hours. The high-speed solar wind streams do not play a significant role in causing the LAWE events on short-term basis, however it may be responsible in causing these events on long-term basis (Mishra and Mishra 2007). Occurrence of LAWE is dominant, when the polarity of Bx and Bz remains positive and polarity of By remains negative, which is never been reported earlier. The amplitude of first harmonic shows good anti-correlation and direction of first and third harmonic shows nearly good anti-correlation with solar wind velocity, whereas the direction of second harmonic shows nearly good anti-correlation with interplanetary magnetic field strength.