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How does one find non-trivial solutions $\vec{x}$ of the following beast

$$\sum_{j=1}^N \sum_{k=1}^N \sum_{l=1}^N M_{ijkl} x_j x_k x_l = 0$$

All numbers are real. Also of interest is the name of this problem, if it has a name. An analogous problem in 2D would be to find the null-space of a matrix. I would naively assume that the name of this problem is to find the null-space of a rank-4 tensor. Is that right?

The origin of the problem is a finite element method discretization of a non-linear ODE.

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