dimension at most $1$ (Schur’s Lemma, see [@Lang2002 Chap. XVII, Proposition 1.1] and [@Newman1967 Lemma 2] for a matrix version of this lemma). After putting the equations in triangular form, randomly sampling over the free variables an element in $V$ yields, thanks to Schwartz-Zippel-DeMillo-Lipton Lemma ([@DL78; @Z79]), a solution to [$D$-`OSMC`]{}with overwhelming probability as soon as ${{\mathbb}{K}}$ is big enough. If one accepts to have a solution matrix over an extension field, we can amplify the probability of success by considering a bigger algebraic extension (see [@Pazzis2010]). Whilst a rather “easy” randomized polynomial-time algorithm solves [`SMC`]{}, the task of finding a deterministic algorithm is more delicate. In our particular case, we can adapt the result of [@DBLP:conf/issac/ChistovIK97] and provide a deterministic polynomial-time algorithm for solving .

### Characteristic $2$ {#characteristic-2 .unnumbered}

Let us recall that in characteristic $2$, the associated matrices $H_1,\ldots,H_m,H_1',\ldots,H_m'$ to quadratic forms can be chosen as upper triangular. In this context, we show in Section \[ss:char\_2\] that [`IP1S`]{}can still be reduced to a $(H_1+H_1^{\operatorname{T}})$-conjugacy problem. Under certain conditions in even dimension, we can solve this conjugacy problem in polynomial-time. These results are well confirmed by some experimental results presented in Section \[ss:bench\]. We can recover a solution 