While Lej da San Murezzan forms the focal point of the town and the path around it is a great starter running route, it is far from the only beautiful lake or flat running route in the area. For a whole new world of beauty and a great trail network, head roughly south west from St Moritz Bad along the edge of the river En valley, towards Champfèr.
As you approach Champfèr, you'll reach a footbridge across the river. You can choose to go left of the river from here, along the edge of the forest, or take the more open route on the Champfèr side of the river (or better still, take one route out and the other back. The route on the Champfèr side is marginally longer, and also wider and more level, though the route on the forest side is also very runnable, but slightly rocky as you reach Silvalpana/Surlej.
When you reach the bridge between Silvaplana/Surlej, cross the bridge, and take the path to the left (Surlej side) of Lej da Silvaplauna. There is a path at the Silvaplana side, which runs along by the kitesurfing and windsurfing centres, but ends at the campsite. The first section of the path is through open grassland, slightly above the level of the lake, but as it meets the forest it rejoins the lake edge. On a calm sunny morning, the vistas from here are particularly spectacular. On a windy Saturday afternoon, the lake will be a blaze of colour as windsurfers and kitesurfers get their weekly adrenaline rush.
At the far side of the lake, you'll reach Sils im Edigan/Segl. If you are going for a really long run, you can carry on through Sils/Segl and run alongside Lej da Segl to Maloja. Those of us with purely human qualities, however, will likely have reach our limits long before this point, and will have turned and headed back towards St Moritz - I have not comments to make on this section of the run; I simply didn't make it that far.
It's approximately 10km from St Moritz to Sils, and a further 6km to the far side of Lej da Segl. This is a mostly flat route, though you can add hills if you please - the forests to the south of the lakes provide plenty of opportunity to do so.