Excellent question ( ... meaning I had to look it up and figure it out because I did not know the answer in my head ...
).
From the IRC. (underlining and bold are mine)
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R311.5.6 Handrails.Handrails shall be provided on at least
one side of each continuous run of treads or flight with four
or more risers.
A flight of stairs is a continuous run of treads and risers, so that's a bit of redundancy there.
Thus, if there is a flight of stairs which has four or more risers, which could be three treads and two landings (one landing at each end of the flight) as that makes four risers, there needs to be a handrail.
Your photo shows three flights of stairs, the lower flight, the upper flight and the center flight (one stair) between the two intermediate landings.
The lower flight has more than 4 risers, so it needs a handrail.
The upper flight has more than 4 risers, so it needs a handrail.
The center flight has 2 risers, so it does not need a handrail.
Now, if those two landings were winders instead of landings, then there would be one continuous flight of stairs, and the handrail would need to be continuous from top to bottom.
Again, though, those brackets, and that fancy spamancy plate around the support brackets are not acceptable.
How much overhang is there on the cap on that center wall? More than 4-1/2"?
Weird base treatment at those landings.
The space between the upper handrail and the wall looks correct, the space between the lower handrail and the wall looks like it makes the handrail project out too far from the wall (more than 4-1/2").