The same storm that brought 6+ inches of rainfall and flash flooding to California as well as 6+ feet to the Sierra Nevada late the previous week moved into the Great Lakes on January 31st and February 1st. This system joined with another system to develop a coastal low off the NJ coast during the night of February 1st and slowly moved across the Gulf of Maine on February 2nd.  A Nor'easter with snowfall of 18 inches in NYC and more than 2 feet in parts of Northern NJ/Eastern PA during the 31st and 1st slowly spread snow northward into northern NY.
Snowfall developed just after midnight on February 2nd across northern New York with a snow band remaining nearly stationary near the Canadian border through midday. As secondary band of snow, with snowfall rates of 1-2+ inches an hour moved into the northern Champlain Valley of NY during the later afternoon and early evening. Snowfall amounts ranged from 4 to 6 inches in the southern Champlain Valley, and southern St. Lawrence valley to 12-18 inches in the northern Adirondacks.

Winter Storm in WESTERN CLINTON, NEW YORK
From 02-FEB-21 05:00:00 to 03-FEB-21 01:00:00 (EST-5)
Property damage: 10.00K
Crop damage: 0.00K
Reported by: CoCoRaHS

Snowfall totals ranged from 8 to 12 inches in the valley of eastern Clinton with 10 to 18 inches in the slopes of western Clinton county.

Winter Storm in SOUTHERN FRANKLIN, NEW YORK
From 02-FEB-21 06:00:00 to 03-FEB-21 00:00:00 (EST-5)
Property damage: 10.00K
Crop damage: 0.00K
Reported by: CoCoRaHS

Snowfall totals ranged from 6 to 12 inches across southern Franklin county.

Winter Weather in EASTERN ESSEX, NEW YORK
From 02-FEB-21 03:00:00 to 02-FEB-21 21:00:00 (EST-5)
Property damage: 5.00K
Crop damage: 0.00K
Reported by: CoCoRaHS

Snowfall totals across eastern Essex county were 4 to 8 inches.

Winter Storm in SOUTHEASTERN ST. LAWRENCE, NEW YORK
From 02-FEB-21 08:00:00 to 03-FEB-21 00:00:00 (EST-5)
Property damage: 10.00K
Crop damage: 0.00K
Reported by: CoCoRaHS

Snowfall totals ranged from 8 to 12 inches across southeast St. Lawrence county.

Winter Storm in NORTHERN FRANKLIN, NEW YORK
From 02-FEB-21 08:00:00 to 03-FEB-21 01:00:00 (EST-5)
Property damage: 10.00K
Crop damage: 0.00K
Reported by: CoCoRaHS

Snowfall totals ranged 10 to 18 inches across northern Franklin county.

Winter Storm in EASTERN CLINTON, NEW YORK
From 02-FEB-21 05:00:00 to 02-FEB-21 22:00:00 (EST-5)
Property damage: 10.00K
Crop damage: 0.00K
Reported by: CoCoRaHS

Snowfall totals ranged from 8 to 12 inches in the valley of eastern Clinton with 10 to 18 inches in the slopes of western Clinton county.

Winter Storm in WESTERN ESSEX, NEW YORK
From 02-FEB-21 06:00:00 to 03-FEB-21 00:00:00 (EST-5)
Property damage: 10.00K
Crop damage: 0.00K
Reported by: CoCoRaHS

Snowfall totals ranged from 6 to 12 inches across western Essex county.

Winter Weather in SOUTHWESTERN ST. LAWRENCE, NEW YORK
From 02-FEB-21 09:00:00 to 02-FEB-21 23:00:00 (EST-5)
Property damage: 5.00K
Crop damage: 0.00K
Reported by: CoCoRaHS

Snowfall totals across the St. Lawrence valley were 6 to 10 inches.

Winter Storm in NORTHERN ST. LAWRENCE, NEW YORK
From 02-FEB-21 09:00:00 to 03-FEB-21 01:00:00 (EST-5)
Property damage: 10.00K
Crop damage: 0.00K
Reported by: CoCoRaHS

Snowfall totals ranged from 8 to 14 inches in northern St. Lawrence county.