The central issue for the jury is the role psychosis played in the mind of Richard Kachkar, 46, when he killed Sgt. Ryan Russell, 35, two years ago.
Jurors deliberated until nearly 9 p.m. Tuesday after considering the case for about five hours on Monday.
The judge has said there is no doubt that Kachkar was the man who stole the plow and hit Russell, but what they must weigh is his mental state at the time.
The defence has urged the jury to find Kachkar not criminally responsible on account of mental disorder, while the Crown has argued for a first-degree murder conviction.
The judge has told the jury they should not be concerned with the consequences of their verdict, but he assured them that not criminally responsible doesn't mean Kachkar would immediately return to the community.
People who are found not criminally responsible are sent to mental health facilities for an indeterminate period of time and can be released only when a review board finds they are no longer a significant threat to public safety.
Court has heard Kachkar, who pleaded not guilty to first-degree murder, yelled about the Taliban, Chinese technology and said "it's all a Russian video game" while driving around the city with the stolen plow.