Solving linear systems of equations modulo a prime power

Question

I am interested in knowing if there is a general theory for solving linear systems of equations modulo a prime power $p^k$. Such questions have been asked here before:

• Solving system of linear equations involving modulo power of 2
• System of linear congruences modulo a prime power

but none of them is satisfactory since the first one only points to other not-exactly-relevant questions, and the second points to Gaussian elimination, which AFAIK only gets as general as working over fields.

I think that a reasonable and generic way of finding such solutions is by using a generalization of Hensel's lemma to multivariate equations, but before I dive into such approach I would like to know if there is not an off-the-shelf solution I could use.

Thanks!