This is a very rare incomplete injury and usually is caused by a medical mishap such as a tumour or lack of blood supply. The only part of the cord to be damaged are the posterior nerve pathways.
There is good voluntary muscle power and they can feel hot or cold, sharp or blunt but they have no light touch, vibration or joint position sense. They therefore find it very difficult to control movement, as they cannot feel the ground beneath their feet or they cannot tell how hard to hold a delicate object etc.