How is the Solarian League Navy organized?
There is a Solarian League Navy. There are also smaller naval forces (usually referred to as, for example, the Beowulf Self-Defense Force) which answer to the local government(s) who raised them. Technically, all of them would come under SLN command in the event of a war or a declared state of emergency; actually, many (if not most) of them would first listen very carefully to what the governments who paid for the ships and provided the personnel to crew them want done. The SDFs (self-defense forces) provide the bulk of the light forces required to prevent outbreaks of piracy in their own areas, and may be tasked by the SLN (with the local governments' concurrence) to extend their activities to other systems in their area if those other systems can't/won't put up the cash to build SDFs of their own to do the job. The SLN has a higher percentage of light units than the SKM or the PN because it is charged with supporting the gendarmerie and other bureaucratic organs operating along the fringes of the League and because whenever anti-piracy operations outside League territory are required, the SLN is responsible for carrying them out.
The SLN's capital ships tend to be older than those of most other navies. This is largely because no one has been stupid enough to attack them for the last several centuries. Remember that prior to the current Manticore-Havenite war, warfighting technology had plateaued over a relatively static period of centuries. As a result, the SLN has had a tendency to retain older units with periodic refits to update their systems and such refits have tended to be relatively minor and system-specific, not general in nature. Although the Sollies' active fleet strength is several times that of the prewar PN, they also maintain a massive reserve of older, mothballed ships which would at least double their tonnage if activated. Unfortunately, the reserve fleet has not been updated in a long time. The theory (officially) is that there is little point in updating them until they are actually needed, since whatever we do to them now will be dated by the time we actually need them, so we might as well modernize them fully then rather than throw money down a rathole by doing partial modernizations (which will only have to be redone anyway) when the time comes. In actuality, however, the reserve is a source of false confidence for the SLN. Even people who know better intellectually tend to look at them as effective warships which can be called quickly to action without fully considering the cost, yard time, and manning (i.e., crew training time) requirements.