Murray tendered his resignation following Saturday's 3-2 home defeat at the hands of Brackley Town — and it has subsequently been accepted by the United board.
Newton (pictured) commented: "Adam has worked extremely hard during his time at Boston United, not only with his commitment to the first-team, but also the club as a whole.
"It is very sad that his dedication has not resulted in the progression of the first-team that we all hoped for and we reluctantly accepted his resignation.
"Adam is a hard-working and professional man and we wish him and his family all the very best for the future.
"The search for a new manager will now commence.
"Karl Hawley will take charge of the first-team until such time as a new manager is appointed and we hope the fans will support Karl through this transitional period."
Applications for the managerial vacancy are now being accepted. CVs and covering letters can be submitted to board@bufc.co.uk in the first instance.
The closing date for applications is 5pm on Friday, November 3rd.