Find all possible ordinals $\alpha,\beta$ satisfying the equations:
$\alpha+\beta=\omega$
$\alpha+\beta=\omega^2+1$
For 1), discarding the trivial cases $\alpha=0$ or $\beta=0$, we have that $\omega^{\beta}\neq \omega$ for all $\beta\neq 1$, and $\omega\cdot k\neq\omega$ for all $k\neq 1$ (I guess these are obvious statements?) Also, $\omega+n\neq\omega$ (for $n>0$) and $n+\omega=\omega$ (for $n\geq 0$). Therefore, the only possibilities (taking into account the normal form of $\alpha$ and $\beta$ are $\alpha=n\in\omega$ and $\beta=\omega$, or $\alpha=\omega$ and $\beta=0$. I think these are all the solutions, since ordinals can't cancel each other to $0$, but I do not know if this answer is completely justified.
For 2) I did the same. Discarding $\alpha=0$ or $\beta=0$: If $\alpha=n\in\omega$, then necessarily $\beta=\omega^2+1$, otherwise the equation is not satisfied. If $\alpha$ is infinite, then necessarily $\alpha=\omega^2$, otherwise other powers of $\omega$ would appear, and $\beta$ cannot cancel them out. Then $\beta=1$, otherwise the equation is not satisfied.
Is this enough?
Recalling that ordinal addition is not commutative, such that $\alpha$ is "absorbed/annihilated" by $\beta > \alpha$ whenever $\beta \ge \omega$, i.e. $\alpha + \beta = \beta$ in that case $-$ it is due to the fact that the addition is interpreted as iterations of the successor function, with limit ordinals treated as unavoidable steps.
In consequence, the first equation $\alpha + \beta = \omega$ admits the following solutions $$ \left\{ \begin{array}{l} \alpha = \omega \\ \beta = 0 \end{array} \right. \quad\&\quad \left\{ \begin{array}{l} \alpha = n \\ \beta = \omega \end{array} \right. \quad\mathrm{with}\, n \in \mathbb{N}, $$ while the second equation $\alpha + \beta = \omega^2 + 1$ leads to $$ \left\{ \begin{array}{l} \alpha = \omega^2 + 1 \\ \beta = 0 \end{array} \right. ,\quad \left\{ \begin{array}{l} \alpha = \omega^2 \\ \beta = 1 \end{array} \right. \quad\&\quad \left\{ \begin{array}{l} \alpha = \omega \cdot n + m \\ \beta = \omega^2 + 1 \end{array} \right. \quad\mathrm{with}\, n,m \in \mathbb{N}. $$