How to find the base point given public and private key and EC parameters except the base point I'm working on the curve secp256k1 and I've private key $k$ and public key $P = [k]G$. I look for solutions to the following problems;

What is the inverse of a scalar $k$ in ECC?

If I'm only missing $G$ but know all parameters of the curve can I find $G$ from $k$ and $P$?
Most standardized curves has a base point $G$ of prime order $q$ so computing the inverse of $k$ (the private key) modulo $q$ can be easily done as
$$
k_{\text{inv}} = k^{p-2} \bmod q,
$$
thanks to Fermat's Little Theorem since
$$
k_{\text{inv}}\times k \equiv k^{p-1} \equiv 1 \pmod{q}
$$
In the case the order is a composite integer $n$, then Euclid Extended algorithm is a good way to find it.
In your problem you have a public point $P_{\text{pub}}$ such that
$$
P_{\text{pub}} = [k]G,
$$
and you only know $k$ and $P_{\text{pub}}$. To find $G$, use of the above algorithm (if the order of $G$ is known) to find $k_{\text{inv}}$ and we have
$$
G = [k_{\text{inv}}] P_{\text{pub}}.
$$