As a balalaika player I would offer just this, the balalaika shape does present problems in holding.It is traditionally played sitting because the neck is not "held" as the thumb frets the two EE strings and the four fingers fret the melody A string. This means that the balalaika is held in a strange way , with the base resting in the inside of the right hand thigh ,the neck is supported by the web of the thumb and palm but loosely and not gripped.
This would not be a huge problem for the uker because the fretting arrangement is different...but then as a ukulele player I feel that then you would need to strap it and would only really be able to play it stood up a la flying vee....You can't rest them on your knee as the point will dig in.
To hold it in the traditional Balalaika way you are advised to put some sort of grippy material ,like felt or a velcro on the base which will hold to your trouser material.....