Solve the problem.

$12)$ Jack's birth date in numbers $($month$-$day$-$year$)$ is a $6-$digit palindrome, say satisfying these conditions: a $+$ b $!=$ c $+$ d; b $+$ e $=($c $+$ d$)+10$;

he was born later than $1958$;

had he been born $1$ month later, his birthday would fall on the same day as his mother's birthday; his mother was born on an even numbered day after the $16$th of the month;

in the year $2023,$ Jack will be in his $50\text{'}$s$.$

On what month and day was his mother born?