It would be nice to define the drawing info on the service side, so that all Clients are using the same drawing info.
Hello Jan -
No, your options for changing the symbology for a stream layer in a web map to render the data being broadcast from a stream service are limited. You cannot use attributes to drive unique value rendering. Pretty much you are limited to a single symbol, although that symbol does not need to be a simple marker symbol, it can be an image (PNG, GIF, or JPEG).
See Also: How create symbology on stream layer
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