In this paper, we present necessary and sufficient conditions of boundedness of $\mathbb{L}$-index in joint variables
for vector-functions analytic in the unit ball, where
$\mathbf{L}=(l_1,l_2): \mathbb{B}^2\to\mathbb{R}^2_+$ is a positive continuous vector-function,
$\mathbb{B}^2=\{z\in\mathbb{C}^2: |z|=\sqrt{|z_1|^2+|z_2|^2}\le 1\}.$
Particularly, we deduce analog of Fricke's theorems for this function class, give estimate of maximum modulus on the skeleton of bidisc.
The first theorem concerns sufficient conditions. In this theorem we assume existence of some radii,
for which the maximum of norm of vector-function on the skeleton of bidisc with larger radius does not exceed maximum of norm of
vector-function on the skeleton of bidisc with lesser radius multiplied by some costant depending only on these radii.
In the second theorem we show that boundedness of $\mathbf{L}$-index in joint variables implies validity of the mentioned estimate for all radii.
ON EXISTENCE OF MAIN POLYNOMIAL FOR ANALYTIC VECTOR-VALUED FUNCTIONS OF BOUNDED L-INDEX IN THE UNIT BALL