I want to get first day of every corresponding month of current year. For example, if user selects \'2010-06-15\', query demands to run from \'2010-06-01\' instead of \'2010
You can use EXTRACT to get the date parts you want:
EXTRACT
EXTRACT( YEAR_MONTH FROM DATE('2011-09-28') ) -- 201109
This works well for grouping.